morfismos, vol 16, no 1, 2012

VOLUMEN 16NÚMERO 1

ENERO A JUNIO DE 2012ISSN: 1870-6525

MorfismosDepartamento de Matematicas

Cinvestav

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VOLUMEN 16NÚMERO 1

ENERO A JUNIO DE 2012ISSN: 1870-6525

MorfismosDepartamento de Matematicas

Cinvestav

Morfismos, Volumen 16, Numero 1, enero a junio de 2012, es una publicacionsemestral editada por el Centro de Investigacion y de Estudios Avanzadosdel Instituto Politecnico Nacional (Cinvestav), a traves del Departamentode Matematicas. Av. Instituto Politecnico Nacional No. 2508, Col. San PedroZacatenco, Delegacion Gustavo A. Madero, C.P. 07360, D.F., Tel. 55-57473800,www.cinvestav.mx, [email protected], Editores Generales: Drs.Isidoro Gitler Golwain y Jesus Gonzalez Espino Barros. Reserva de DerechosNo. 04-2012-011011542900-102, ISSN: 1870-6525, ambos otorgados por elInstituto Nacional del Derecho de Autor. Certificado de Licitud de TıtuloNo. 14729, Certificado de Licitud de Contenido No. 12302, ambos otorga-dos por la Comision Calificadora de Publicaciones y Revistas Ilustradas de laSecretarıa de Gobernacion. Impreso por el Departamento de Matematicas delCinvestav, Avenida Instituto Politecnico Nacional 2508, Colonia San PedroZacatenco, C.P. 07360, Mexico, D.F. Este numero se termino de imprimir enseptiembre de 2012 con un tiraje de 50 ejemplares.

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Morfismos

Contents - Contenido

The Mathematical Life of Nikolai Vasilevski

Sergei Grudsky . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Sistemas de funciones iteradas por partes

Sergio Mabel Juarez Vazquez y Flor de Marıa Correa Romero . . . . . . . . . . . . 9

Uniqueness of the Solution of the Yule–Walker Equations: A Vector SpaceApproach

Ana Paula Isais-Torres and Rolando Cavazos-Cadena . . . . . . . . . . . . . . . . . . . 29

Morfismos, Vol. 16, No. 1, 2012, pp. 1–8

The Mathematical Life of

Nikolai Vasilevski

Sergei Grudsky

2010 Mathematics Subject Classification: 30C40, 46E22, 47A25, 47B10,47B35, 47C15,47L15, 81S10.Keywords and phrases: C∗-algebras, multidimensional singular integral op-erators, Toeplitz-Bergman operators, Toeplitz-Fock operators, Berezin quan-tization procedure.

Nikolai Vasilevski’s creative attitude — not only to mathematics, butto life in general — had already received a great stimulus in Odessa HighSchool 116. This highly selective high school accepted talented childrenfrom all over the city, being famous for its selection of quality teachers. Acreative, nonstandard, yet highly personal approach to teaching combinedwith a demanding attitude towards students. The school was also famousfor its system of self-government by the students, quite unusual by tradi-tional Soviet standards. Many students that graduated from this schoollater became well-known scientists and productive researchers.

Upon graduation in 1966, Nikolai became a student at the Departmentof Mathematics and Mechanics of Odessa State University. In his third yearof studies he began his serious mathematical work under the supervisionof the well known Soviet mathematician Georgiy Semenovich Litvinchuk.Litvinchuk was a gifted teacher and scientific adviser, with a talent forfascinating his students with problems long interesting and yet up-to-date.The weekly Odessa seminar on boundary value problems, chaired by Prof.Litvinchuk for more than 25 years, deeply influenced Nikolai Vasilevski aswell as other students.

Thus Nikolai started to work on the problem of developing Fredholmtheory for a class of integral operators with nonintegrable integral kernels.In essence, the integral kernel was the Cauchy kernel multiplied by a loga-rithmic factor. Integral operators of this type lie in between singular inte-gral operators and those whose kernels have weak (integrable) singularities.A famous Soviet mathematician, F. D. Gakhov, posed this problem in the

1


The Mathematical Life of

Nikolai Vasilevski

Sergei Grudsky

2010 Mathematics Subject Classification: 30C40, 46E22, 47A25, 47B10,47B35, 47C15,47L15, 81S10.Keywords and phrases: C∗-algebras, multidimensional singular integral op-erators, Toeplitz-Bergman operators, Toeplitz-Fock operators, Berezin quan-tization procedure.

Nikolai Vasilevski’s creative attitude — not only to mathematics, butto life in general — had already received a great stimulus in Odessa HighSchool 116. This highly selective high school accepted talented childrenfrom all over the city, being famous for its selection of quality teachers. Acreative, nonstandard, yet highly personal approach to teaching combinedwith a demanding attitude towards students. The school was also famousfor its system of self-government by the students, quite unusual by tradi-tional Soviet standards. Many students that graduated from this schoollater became well-known scientists and productive researchers.

Upon graduation in 1966, Nikolai became a student at the Departmentof Mathematics and Mechanics of Odessa State University. In his third yearof studies he began his serious mathematical work under the supervisionof the well known Soviet mathematician Georgiy Semenovich Litvinchuk.Litvinchuk was a gifted teacher and scientific adviser, with a talent forfascinating his students with problems long interesting and yet up-to-date.The weekly Odessa seminar on boundary value problems, chaired by Prof.Litvinchuk for more than 25 years, deeply influenced Nikolai Vasilevski aswell as other students.

Thus Nikolai started to work on the problem of developing Fredholmtheory for a class of integral operators with nonintegrable integral kernels.In essence, the integral kernel was the Cauchy kernel multiplied by a loga-rithmic factor. Integral operators of this type lie in between singular inte-gral operators and those whose kernels have weak (integrable) singularities.A famous Soviet mathematician, F. D. Gakhov, posed this problem in the

1

2 Sergei Grudsky

early 1950s, and it remained open for more than 20 years. Nikolai man-aged to provide a complete solution in a much more general setting thanthe original. While working on this problem, Nikolai revealed a key traitof his mathematical talent: his ability to penetrate deeply into the core ofthe problem, and to see rather unexpected connections between differenttheories. For instance, in order to solve Gakhov’s problem, Nikolai utilizedthe theory of singular integral operators with coefficients having disconti-nuities of first kind, and the theory of operators whose integral kernels havefixed singularities — both theories having just appeared at that time. Thesuccess of the young mathematician was well recognized by a broad circleof experts working in the area of boundary value problems and operatortheory. Nikolai was awarded the prestigious M. Ostrovskii Prize in 1971,given to young Ukrainian scientists with the best research work. Due tohis solution of the famous problem, Nikolai quickly entered the mathemat-ical community, and became known to many prominent mathematiciansof that time. In particular, he was influenced by regular interaction withoutstanding mathematicians such as M. G. Krein and S. G. Mikhlin.

Nikolai graduated from Odessa State University in 1971, obtaining hisMaster degree. After two years he defended his Ph.D. thesis, and in thesame year he became an Assistant Professor at the Department of Mathe-matics and Mechanics of Odessa State University, where he was later pro-moted first to the rank of Associate Professor and then to Full Professor.

Having received the degree, Nikolai continued his active mathematicalwork. He quickly displayed yet another facet of his talent in approach-ing mathematical problems: his vision and ability to use general algebraicstructures in operator theory, which, on the one hand, simplify the prob-lem, and on the other, can be applied to many different problems. We willbriefly describe two examples of this.

The first example is the method of orthogonal projections. In 1979,studying the algebra of operators generated by the Bergman projectiontogether with the operators of multiplication by piecewise continuous func-tions, N. Vasilevski gave a description of the C∗-algebra generated by twoself-adjoint elements s and n satisfying the properties s2 + n2 = e andsn+ns = 0. A simple substitution p = (e+ s−n)/2 and q = (e− s−n)/2shows that this algebra is also generated by two self-adjoint idempotents(orthogonal projections) p and q (and the identity element e). During thelast quarter of the past century, the latter algebra was rediscovered by manyauthors around the world. Among all algebras generated by orthogonal pro-jections, the algebra generated by two projections is the only tame algebra(excluding the trivial case of the algebra with identity generated by oneorthogonal projection). All algebras generated by three or more orthogonal

The Mathematical Life of Nikolai Vasilevski 3

projections are known to be wild, even when the projections satisfy someadditional constraints. Many model algebras arising in operator theoryare generated by orthogonal projections, and thus any information abouttheir structure essentially broadens the set of operator algebras admitting areasonable description. In particular, two and more orthogonal projectionsnaturally appear in the study of various algebras generated by the Bergmanprojection and by piecewise continuous functions having two or more dis-tinct limiting values at a point. Although these projections, say, P , Q1,. . . , Qn, satisfy the extra condition Q1 + · · ·+Qn = I, they still generate,in general, a wild C∗-algebra. However, it was shown that the structure ofthis algebra is determined by the joint properties of certain positive injec-tive contractions Ck, k = 1, . . . , n, satisfying the identity

∑nk=1 Ck = I; the

structure is therefore determined by the structure of the C∗-algebra gener-ated by the contractions. The principal difference between the case of twoprojections and the general case of a finite set of projections is now com-pletely clear: for n = 2 (with projections P and Q+ (I −Q) = I) we haveonly one contraction, and the spectral theorem leads directly to the desireddescription of the algebra. For n > 2 we have to deal with the C∗-algebragenerated by a finite set of noncommuting positive injective contractions,which is a wild problem. Fortunately, for many important cases relatedto concrete operator algebras these projections have yet another specialproperty: the operators PQ1P, . . . , PQnP commute. This property makesthe respective algebra tame, so it has a nice and simple description as thealgebra of all n×n matrix valued functions that are continuous on the jointspectrum ∆ of the operators PQ1P, . . . , PQnP , with a certain degenerationon the boundary of ∆.

Another notable example of the algebraic structures used and devel-oped by N. Vasilevski is his version of the Local Principle. The notionsof locally equivalent operators and localization theory were introduced anddeveloped by I. Simonenko in the mid-sixties. According to the traditionof that time, the theory was focused on the study of individual operators,and on the reduction of the Fredholm properties of an operator to localinvertibility. Later, different versions of the local principle were elaboratedby many authors, including G. R. Allan, R. Douglas, I. Ts. Gohberg, N. Ia.Krupnik, A. Kozak, and B. Silbermann. In spite of the fact that many ofthese versions are formulated in terms of Banach- or C∗-algebras, the mainresult, as before, reduces invertibility (or the Fredholm property) to localinvertibility. On the other hand, at about the same time, several papers onthe description of algebras and rings in terms of continuous sections werepublished by J. Dauns and K. H. Hofmann, M. J. Dupre, J. M. G. Fell,M. Takesaki and J. Tomiyama. These two directions have been developedindependently, with no known links between the two series of papers. N.Vasilevski was the one who proposed a local principle which gives the global

4 Sergei Grudsky

description of the algebra under study in terms of continuous sections of acertain canonically defined C∗-bundle. This approach is based on generalconstructions of J. Dauns and K. H. Hofmann, and results of J. Varela.The main contribution consists of a deep re-comprehension of the tradi-tional approach to the local principles unifying the ideas coming from bothdirections, resulting in a canonical procedure that provides the global de-scription of the algebra under consideration in terms of continuous sectionsof a C∗-bundle constructed by means of local algebras.

In the eighties and even later, the main direction of the work of NikolaiVasilevski has been the study of multidimensional singular integral opera-tors with discontinuous coefficients. The main philosophy here is first tostudy algebras containing these operators, thus providing a solid foundationfor the study of various properties (in particular, the Fredholm property)of concrete operators. The main tool has been the version of the local prin-ciple described above. This principle was not merely used to reduce theFredholm property to local invertibility but also for a global description ofthe algebra as a whole based on the description of the local algebras. Usingthis methodology, Nikolai Vasilevski obtained deep results in the theoryof operators with Bergman’s kernel and piece-wise continuous coefficients,the theory of multidimensional Toeplitz operators with pseudodifferentialpresymbols, the theory of multidimensional Bitsadze operators, the theoryof multidimension al operators with shift, etc. N. Vasilevski defended theDoctor of Sciences dissertation in 1988, based on these results, entitled“Multidimensional singular integral operators with discontinuous classicalsymbols”.

Besides being a very active mathematician, N. Vasilevski is an excellentlecturer. His talks are always clear, sparkling, and full of humor, so naturalfor someone who grew up in Odessa, a city with a longstanding traditionof humor and fun. He was the first at Odessa State University to designand teach a course in general topology. Students enjoyed his lectures incalculus, real analysis, complex analysis, and functional analysis. He wasone of the most popular professors at the Department of Mathematics andMechanics of Odessa State University. Nikolai is a master of presentations,and his colleagues always enjoy his talks at conferences and seminars.

In 1992 Nikolai Vasilevski moved to Mexico. He started his career thereas an Investigator (Full Professor) at the Mathematics Department of theCinvestav (Centro de Investigacion y de Estudios Avanzados). His ap-pointment significantly strengthened the Department, which is one of theleading mathematical centers in Mexico. His relocation also visibly revital-ized mathematical activity in the country in the field of operator theory.While actively pursuing his own research agenda, Nikolai also served as theorganizer of several important conferences. For instance, we mention the


(regular since 1998) annual workshop “Analisis: Norte-Sur,” and the well-known international conference IWOTA-2009. He facilitated the relocationto Mexico of a number of other experts in operator theory, including Yu.Karlovich and S. Grudsky.

During his career in Mexico, Vasilevski produced a sizable group ofstudents and younger colleagues; six young mathematicians have receiveda Ph.D. degree under his supervision.

During his life in Mexico, Vasilevski’s scientific interests concentratedmainly around the theory of Toeplitz operators on Bergman and Fockspaces. At the end of the 1990s, N. Vasilevski discovered a quite sur-prising phenomenon in the theory of Toeplitz operators on the Bergmanspace. Unexpectedly, there exists a rich family of commutative C∗-algebrasgenerated by Toeplitz operators with non-trivial defining symbols. In 1995B. Korenblum and K. Zhu proved that the Toeplitz operators with radialdefining symbols acting on the Bergman space over the unit disk can bediagonalized with respect to the standard monomial basis in the Bergmanspace. The C∗-algebra generated by such Toeplitz operators is thereforeobviously commutative. Four years later Vasilevski also proved the com-mutativity of the C∗-algebra generated by the Toeplitz operators actingon the Bergman space over the upper half-plane and with defining sym-bols depending only on Im z. Furthermore, he discovered the existence of arich family of commutative C∗-algebras of Toeplitz operators. Moreover, itturned out that the smoothness properties of the symbols do not play anyrole in commutativity: the symbols can be merely measurable. Surpris-ingly, everything is governed by the geometry of the underlying manifold,the unit disk equipped with the hyperbolic metric. The precise descrip-tion of this phenomenon is as follows. Each pencil of hyperbolic geodesicsdetermines the set of symbols which are constant on the correspondingcycles, the orthogonal trajectories to geodesics forming the pencil. TheC∗-algebra generated by the Toeplitz operators with such defining symbolsis commutative. An important feature of such algebras is that they remaincommutative for the Toeplitz operators acting on each of the commonlyconsidered weighted Bergman spaces. Moreover, assuming some naturalconditions on “richness” of the classes of symbols, the following completecharacterization has been obtained: A C∗-algebra generated by the Toeplitzoperators is commutative on each weighted Bergman space if and only ifthe corresponding defining symbols are constant on cycles of some pencilof hyperbolic geodesics. It is also worth mentioning that the proof of thisresult uses the Berezin quantization procedure in an essential way. Apartfrom its own beauty, this result reveals an extremely deep influence of thegeometry of the underlying manifold on the properties of the Toeplitz op-erators over the manifold. In each of the mentioned above cases, when the

6 Sergei Grudsky

algebra is commutative, a certain unitary operator has been constructed.It reduces the corresponding Toeplitz operators to certain multiplicationoperators, which also allows one to describe their representations of spec-tral type. This gives a powerful research tool for the subject, in particular,yielding direct access to the majority of the important properties such asboundedness, compactness, spectral properties, and invariant subspaces ofthe Toeplitz operators under study. This new approach has enabled thesolution of a number of important problems in the theory of Toeplitz oper-ators and related areas.

The results of the research in this direction became a part of the mono-graph “Commutative Algebras of Toeplitz Operators on the Bergman Spa-ce” published by N. Vasilevski in Birkhauser-Verlag in 2008.

The extension of the above result from the unit disk to the unit ball wasrecently done by Nikolai together with his Mexican colleague Raul Quiroga.Geometry again played an essential role in this study. The commutativ-ity properties of Toeplitz operators here are governed by the so-called La-grangian pairs, pairs of orthogonal Lagrangian foliations of the unit ballwith certain distinguished geometrical properties. The leaves of one ofthese foliations always turn out to be the orbits of a maximal abelian sub-group of biholomorphisms of the unit ball. The result says that, given anyLagrangian pair, the C∗-algebra generated by Toeplitz operators, whosegenerating symbols are constant on the leaves being orbits, is commutativeon each commonly considered weighted Bergman space on the unit ball.

The program of studying commutative algebras generated by Toeplitzoperators as well as the development of various related problems, initiatedby N. Vasilevski, is now being carried out by growing groups of mathemati-cians in different research centers.

During his twenty years at the Cinvestav, Nikolai Vasilevski has consis-tently applied the best traditions of the Russian mathematical school in histraining of young talented Mexican researchers. The constantly growinggroup of his coauthors, colleagues, and students is an established part ofthe “Mexican school of Toeplitz operators”—an expression heard more andmore at international conferences.

Selected papers by Nikolai Vasilevski

1. Vasilevski N. L., On a class of singular integral operators with kernelsof polar-logarithmic type, Izvestija Akad. Nauk SSSR ser. matem.40:1 (1976), 131–151 (In Russian). English translation: Math. USSRIzvestija 10:1 (1976), 127–143.

2. Vasilevski N. L., Two-dimensional Mikhlin-Calderon-Zygmund oper-ators and bisingular operators, Sibirski Matematicheski Zurnal 27:2


(1986), 23–31 (In Russian). English translation: Siberian Math. J.27:2 (1986), 161–168.

3. Vasilevski N. L., Algebras generated by multidimensional singular in-tegral operators and by coefficients admitting discontinuities of homo-geneous type, Matematicheski Sbornik 129:1 (1986), 3–19 (In Rus-sian). English translation: Math. USSR Sbornik 57:1 (1987), 1–19.

4. Vasilevski N. L., On an algebra connected with Toeplitz operatorson the tube domains, Izvestija Akad. Nauk SSSR, ser. matem.51:1 (1987), 79–95 (In Russian). English translation: Math. USSRIzvestija 30:1 (1988), 71–87.

5. Vasilevski N. L.; Trujillo R., Convolution operators on standard CR -manifolds. I. Structural Properties, Integral Equations and OperatorTheory 19:1 (1994), 65–107.

6. Vasilevski N. L., Convolution operators on standard CR - manifolds.II. Algebras of convolution operators on the Heisenberg group, IntegralEquations and Operator Theory, 19:3 (1994), 327–348.

7. Vasilevski N. L., C*-algebras generated by orthogonal projections andtheir applications, Integral Equations and Operator Theory 31 (1998),113–132.

8. Vasilevski N. L., On the structure of Bergman and poly-Bergmanspaces, Integral Equations and Operator Theory 33 (1999), 471–488.

9. Vasilevski N. L., Poly-Fock spaces, Operator Theory. Advances andApplications 117 (2000), 371–386.

10. Grudsky S.; Vasilevski N. L., Bergman-Toeplitz operators: radialcomponent influence, Integral Equations and Operator Theory 40:1(2001), 16–33.

11. Vasilevski N. L., Toeplitz operators on the Bergman spaces: Inside-the-Domain effects, Contemporary Mathematics 289 (2001), 79–146.

12. Vasilevski N. L., Bergman space structure, commutative algebras ofToeplitz operators and hyperbolic geometry, Integral Equations andOperator Theory 46 (2003), 235–251.

13. Grudsky S.; Quiroga-Barranco R.; Vasilevski N. L., Commutative C∗-algebras of Toeplitz operators and quantization on the unit disk, J.Functional Analysis 234 (2006), 1–44.

8 Sergei Grudsky

14. Vasilevski N. L., On the Toeplitz operators with piecewise continu-ous symbols on the Bergman space, In:“Modern Operator Theoryand Applications”, Operator Theory: Advances and Applications 170(2007), 229–248.

15. Vasilevski N. L., Poly-Bergman spaces and two-dimensional singularintegral operators, Operator Theory: Advances and Applications 171(2007), 349–359.

16. Quiroga-Barranco R.; Vasilevski N. L., Commutative C∗-algebras ofToeplitz operators on the unit ball, I. Bargmann-type transforms andspectral representations of Toeplitz operators, Integral Equations andOperator Theory 59:3 (2007), 379-419.

17. Quiroga-Barranco R.; Vasilevski N. L., Commutative C∗-algebras ofToeplitz operators on the unit ball, II. Geometry of the level sets ofsymbols, Integral Equations and Operator Theory 60:1 (2008), 89–132.

18. Grudsky S.; Vasilevski N. L., On the structure of the C∗-algebra gen-erated by Toeplitz operators with piece-wise continuous symbols, Com-plex Analysis and Operator Theory 2:4 (2008), 525–548.

19. Grudsky S.; Vasilevski N. L., Anatomy of the C∗-algebra generatedby Toeplitz operators with piece-wise continuous symbols, OperatorTheory: Advances and Applications 190 (2009), 243–265.

20. Vasilevski N. L. Quasi-radial quasi-homogeneous symbols and commu-tative banach algebras of Toeplitz operators, Integral Equations andOperator Theory 66:1 (2010), 141–152.

Sergei GrudskyDepartamento de Matematicas,CINVESTAV del I.P.N.,Aportado Postal 14-740,Mexico D.F, C.P. 07360, [email protected]


Sistemas de funciones iteradas por partes∗

Sergio Mabel Juarez Vazquez 1

Flor de Marıa Correa Romero 2

Resumen

En la actualidad los medios digitales se han convertido en herramien-tas indispensables ya que la mayorıa de la informacion se manejamediante ellos. Las imagenes son una forma imprescindible de infor-macion pero lamentablemente ocupan bastante espacio en la memoriade una computadora y a su vez los dispositivos de almacenamientodigital suelen ser caros. Reducir el espacio que las imagenes ocupanen la memoria de una computadora baja los costos y permite que latransmision de estas sea mas eficiente.El objetivo de este trabajo es presentar una aplicacion del conceptode conjunto fractal a la compresion de imagenes digitales. La tecnicaque se usara para la compresion esta basada en la teorıa matematicadenominada sistemas de funciones iteradas por partes.

2010 Mathematics Subject Classification: 68U10, 65D18.Keywords and phrases: sistemas de funciones iteradas, sistemas de fun-ciones iteradas por partes, conjunto fractal, compresion fractal de imagenes.

1 Introduccion

La informacion que actualmente se maneja es en su mayorıa a traves decomputadoras, la importancia que esta tiene para que los sistemas social yeconomico funcionen, trajo consigo que se este constantemente desarrollan-do herramientas para tratarla. Problemas como el asegurar la informaciony almacenarla de manera digital han tomado relevancia.

∗Este trabajo es parte de la tesis de licenciatura desarrollada por el primer autorbajo la direccion de la segunda autora. La tesis fue presentada en el Departamento deMatematicas de la Escuela Superior de Fısica y Matematicas del Instituto PolitecnicoNacional en noviembre de 2010.

1Becario Conacyt.2Profesora Becaria COFAA, Departamento de Matematicas ESFM-IPN.

9


Sistemas de funciones iteradas por partes∗

Sergio Mabel Juarez Vazquez 1

Flor de Marıa Correa Romero 2

Resumen

En la actualidad los medios digitales se han convertido en herramien-tas indispensables ya que la mayorıa de la informacion se manejamediante ellos. Las imagenes son una forma imprescindible de infor-macion pero lamentablemente ocupan bastante espacio en la memoriade una computadora y a su vez los dispositivos de almacenamientodigital suelen ser caros. Reducir el espacio que las imagenes ocupanen la memoria de una computadora baja los costos y permite que latransmision de estas sea mas eficiente.El objetivo de este trabajo es presentar una aplicacion del conceptode conjunto fractal a la compresion de imagenes digitales. La tecnicaque se usara para la compresion esta basada en la teorıa matematicadenominada sistemas de funciones iteradas por partes.

2010 Mathematics Subject Classification: 68U10, 65D18.Keywords and phrases: sistemas de funciones iteradas, sistemas de fun-ciones iteradas por partes, conjunto fractal, compresion fractal de imagenes.

1 Introduccion

La informacion que actualmente se maneja es en su mayorıa a traves decomputadoras, la importancia que esta tiene para que los sistemas social yeconomico funcionen, trajo consigo que se este constantemente desarrollan-do herramientas para tratarla. Problemas como el asegurar la informaciony almacenarla de manera digital han tomado relevancia.

∗Este trabajo es parte de la tesis de licenciatura desarrollada por el primer autorbajo la direccion de la segunda autora. La tesis fue presentada en el Departamento deMatematicas de la Escuela Superior de Fısica y Matematicas del Instituto PolitecnicoNacional en noviembre de 2010.

1Becario Conacyt.2Profesora Becaria COFAA, Departamento de Matematicas ESFM-IPN.

9

10 Sergio Mabel Juarez Vazquez y Flor de Marıa Correa Romero

En el caso del almacenamiento digital de informacion se han creadometodos para reducir su volumen sin que se afecte el contenido de esta.Las imagenes digitales son una gran fuente de informacion. Quien no ha es-cuchado la frase: una imagen dice mas que mil palabras. Como las imagenessuelen ocupar mucho espacio en la memoria de una computadora, se handesarrollado tecnicas de compresion para ayudar a resolver tal problema.

La compresion de imagenes puede ser con perdida o sin perdida de in-formacion. Los algoritmos que realizan la compresion con perdida, reducenla calidad de la imagen reconstruida despues del proceso de compresion,ası pues esta imagen puede discrepar en relacion a la imagen original endetalles como contornos, formas y colores. Con los algoritmos sin perdida,las imagenes procesadas quedan intactas, pero pagan esta ventaja con unarazon de compresion menor en comparacion a los algoritmos con perdida,por lo tanto se ocupar mas memoria para almacenarlas. Para algunos ca-sos se requiere de comprimir imagenes en donde la perdida sea mınima onula. Ejemplos de esto pueden ser imagenes medicas, imagenes de hue-llas dactilares, imagenes donde haya texto, en general imagenes cuya in-formacion contenida se necesita de forma muy precisa. En otros casos sepuede aprovechar de algunas limitaciones que el ojo humano tiene, de lahabilidad que hemos desarrollado para poder intuir formas, del simbolismoque asociamos a ciertas imagenes o simplemente que algunos detalles de laimagen no tienen importancia para nosotros. Si un ser querido nos muestrauna foto en donde el aparece y en el fondo hay algunas nubes, uno pondraatencion primordialmente en la persona sin darle tanta importancia a laforma que cada nube tiene, ademas las nubes tienen formas aleatorias, asıque si la imagen es procesada mediante una tecnica de compresion y enel proceso se pierde informacion sobre la forma exacta que la nube tieneen la imagen, antes y depues del proceso, no lo notaremos o simplementeno importara, pues nos interesara solo el hecho de que es una nube. Lacompresion fractal es una tecnica de las que se denomina con perdida, laimagen una vez decodificada no es igual que la imagen original.

Este trabajo ilustra la aplicacion que la geometrıa fractal tiene a lacompresion de imagenes digitales. Cada vez que se ocupe la palabra imagense hara referencia a una imagen digital a menos que se mencione lo contrario.A lo largo de las secciones siguientes se dara un metodo para codificar ycomprimir imagenes en escala de grises con ayuda del concepto de conjuntofractal como el atractor de un sistema de funciones iteradas por partes(SFIP). Los autores observamos que en toda la literatura que revisamossobre compresion fractal de imagenes, el concepto de SFIP carece de unadefinicion formal y no se comprueba que se puede inducir una contraccion atraves del sistema. En este artıculo se propone una definicion para los SFIP

Sistemas de funciones iteradas por partes 11

y se demuestra que dado un SFIP lo podemos asociar con una contraccion.

Cabe mencionar que la generalizacion de codificacion de imagenes acolores no es difıcil a partir de la teorıa de codificacion de imagenes enescala de grises.

2 Teorıa basica

Los resultados que se mencionan en esta seccion son conocidos, por lo quesolo se enunciaran. El lector puede consultar las demostraciones en [1] y[13].

Sea (X, d) un espacio metrico, denotaremos por H (X) al conjunto detodos los subconjuntos compactos y no vacıos de (X, d).

Definicion 2.0.1. Sea (X, d) un espacio metrico, sean x ∈ X y A,B ∈H (X). Se define la distancia de x al conjunto A como

d(x,A) := min {d(x, y) | y ∈ A} ,

y la distancia de A a B como

d(A,B) := max {d(x,B) | x ∈ A} .

Proposicion 2.0.2. Sea (X, d) un espacio metrico, y sea

h : H (X)× H (X) −→ R

la funcion definida por ∀ A,B ∈ H (X), h(A,B) := d(A,B) ∨ d(B,A),entonces tenemos que h es una metrica sobre H (X).

Teorema 2.0.3. Sea (X, d) un espacio metrico completo, entonces el es-pacio metrico (H (X), h) es completo.

Definicion 2.0.4. Sean (X1, d1) y (X2, d2) dos espacios metricos. Unafuncion f : (X1, d1) −→ (X2, d2) es una funcion de Lipschitz, si existe unnumero real positivo α tal que

∀ x, y ∈ X, d2 (f(x), f(y)) ≤ α d1 (x, y) ,

al numero α se le llama un factor de Lipschitz de la funcion f . Si se cumpleque 0 ≤ α < 1, entonces f es llamada una contraccion y α un factor decontraccion para f .

Teorema 2.0.5. (Teorema del punto fijo). Si (X, d) es un espaciometrico completo y f : X −→ X una contraccion, con α un factor decontraccion de f , entonces existe un unico xf ∈ X tal que f(xf ) = xf , axf se le llama el punto fijo de la contraccion.


2.1 Sistemas de funciones iteradas

Proposicion 2.1.1. Sea (X, d) un espacio metrico completo y sean paraı ∈ {1, . . . , n} fı : (X, d) −→ (X, d) contracciones, con αı un factor decontraccion para fı. Sea F : (H (X), h) −→ (H (X), h) la funcion definidapor

∀ A ∈ H (X), F (A) =

n⋃ı=1

fı(A),

entonces F es una contraccion y α := max {αı|ı ∈ {1, . . . , n}} es un factorde contraccion para F .

Definicion 2.1.2. Un sistema de funciones iteradas o SFI, consiste de unespacio metrico completo (X, d) y una familia finita de contracciones

{fı : (X, d) −→ (X, d) | ı ∈ {1, . . . , k}} ,

al SFI se le denota por {(X, d) ; f1, f2, . . . , fk} , y se llama un factor decontraccion del SFI al numero α := max {αı | ı ∈ {1, . . . , k}} , donde elnumero αı es un factor de contraccion para fı, ı ∈ {1, . . . , k}.

De acuerdo con la Proposicion 2.1.1, dado un sistemas de funcionesiteradas {(X, d) ; f1, f2, . . . , fk} , se puede definir una contraccion F enel espacio metrico completo (H (X), h), y por el Teorema del punto fijoexiste un unico AF ∈ H (X), tal que este es el punto fijo de la contraccion,el conjunto AF es llamado el conjunto fractal asociado al SFI y a F lacontraccion inducida por el SFI.

Ejemplo. Consideremos el siguiente SFI

{(I2, de) ; f1, f2, f3)

},

donde

de es la distancia euclideana,

I = [0, 1],

f1(x, y) := (1/2x+ 1/4, 1/2y + 1/2),

f2(x, y) := (1/2x, 1/2y),

f3(x, y) := (1/2x+ 1/2, 1/2y).

Si F es la contraccion inducida por el SFI y tomamos el compacto C de R2

como la imagen siguiente.


Figura 1: Imagen que representa al compacto C de R2.

Las figuras que se muestran a continuacion de izquierda a derecha y dearriba hacia abajo son respectivamente los resultados obtenidos para

F (C), F ◦(2)(C) := F (F (C)), . . . , F ◦(6)(C).

Figura 2: Como se puede abservar F ◦(6)(C) consta de 729 copias reducidasdel compacto C.

Notemos que todo elemento de (H (X), h) se puede ver como el atractorde un SFI, pues dado C ∈ (H (X), h), la funcion

φ : (H (X), h) −→ (H (X), h)

definida por, φ(A) := C, para todo A ∈ H (X) es una contraccion y

{(H (X), h) ; φ}

es un SFI, que tiene por atractor a C.


Este tipo de funciones son llamadas funciones de condensacion3 y aun SFI que contenga a una de estas funciones se le llama un sistema defunciones iteradas con condensacion.

Teorema 2.1.3. (Teorema del collage para fractales). Sea (X, d) unespacio metrico completo y sean B ∈ H (X) y ε ∈ R+ dados.

Si {(X, d) ; f1, . . . , fk} es un SFI con un factor de contraccion α y AF

el atractor del SFI tales que

h(B,∪k

i=1fi(B))≤ ε, entonces

h(AF , B) ≤ ε

1− α.

Es decir

h(AF , B) ≤ 1

1− αh(B,∪k

i=1fi(B)),

para todo B ∈ H (X) .

El Teorema del collage dice que para tener un SFI cuyo atractor seasemenjante a un conjunto B ∈ H (X), tenemos que fabricar un conjuntode contracciones {f1, . . . , fk}, tal que la union ( o collage ) de los conjuntosf1(B), . . . , fk(B) este cercano al conjunto B.

Por lo tanto si h(B,∪k

i=1fi(B))≤ ε para un ε lo suficientemente peque-

no podemos sustituir a B por el atractor del SFI. El teorema tambien nosayuda a tener una medida de que tan cerca estara el atractor de un SFIa B sin tener que calcular el atractor, basta solo con estimar la distanciaentre B y ∪k

i=1fi(B).

Ademas la aproximacion del atractor al conjunto B sera mejor cuandomas pequeno sea el factor de contracion del SFI y no depende del numerode contraciones que lo forman.

Si tomamos a (R2, de), con de la metrica euclidiana y trabajamos so-bre el conjunto de todos los subconjuntos compactos no vacıos de R2 paraformar al espacio metrico (H (R2), h) en donde una fotografıa o en gene-ral cualquier imagen es considerada como un compacto de R2, entoncespodrıamos aproximar una fotografıa por un conjunto fractal que sea elatractor de un adecuado SFI y si ademas este SFI consta de pocas contrac-ciones, podemos almacenar las contracciones en lugar de la imagen originaly ası habremos reducido el espacio ocupado por la imagen en la memoriade un sistema digital.

3Una funcion de condensacion es una contraccion y cero es un factor de contraccionpara ella.


Esta fue la idea que abrio la investigacion de la compresion fractal deimagenes.

Existen algunos inconvenientes. Por desgracia el Teorema del collageno proporciona un metodo para encontrar dicho SFI y el SFI cuyo atractoraproxima al compacto en cuestion no necesariamente es unico.

Si tenemos por ejemplo que {(X, d) ; ω1, ω2, . . . , ωk} con k ≥ 2 es unode los sistemas de funciones iteradas que aproxima a el compacto, entoncespodemos escoger dos de las contracciones del SFI, digamos ω1 y ω2 paradefinir una nueva contraccion W1,2 : H (X) −→ H (X) como W1,2(A) :=ω1(A) ∪ ω2(A) para todo A ∈ H (X). Ası tenemos un nuevo SFI

{(H (X), h) ; W1,2,W3 . . . ,Wk}

donde

W1,2(A) = ω1(A) ∪ ω2(A) y ∀ i ∈ {3, . . . , k} , Wi(A) = ωi(A),

el cual tiene el mismo atractor que el SFI original y por tanto se acerca alcompacto que querıamos aproximar del mismo modo, pero este tiene k− 1contracciones a diferencia del primero que poseıa k contracciones. De estaforma podemos definir diferentes SFI que tendrıan el mismo atractor, sinembargo hay un aspecto importante que nos hacen preferir un SFI sobreotro y este es que las contracciones que forman al SFI esten definidas deuna manera simple lo cual nos permite intuir como ser el atractor del SFI.Cabe senalar que el Teorema del collage tampoco proporciona una manerade elegir el mejor SFI, sin embargo todo lo anterior no resta importancia aeste resultado.

3 Sistemas de funciones iteradas por partes

El Teorema del collage asegura que dada cualquier imagen, siempre existeun fractal que se le parece tanto como nosotros queramos, pero supongamosque tenemos la fotografıa de uno de nuestros seres queridos y que conocemosun SFI cuyo atractor aproxima a la fotografıa, si iteramos la contraccion in-ducida por el SFI evaluada en la imagen de un arbol, el atractor se pareceraa la fotografıa que estamos intentando aproximar pero como esta compuestade copias transformadas de un arbol, la foto a detalle exhibirıa pequenosarbolitos distorsionados, en particular en los rasgos fısicos de la personalo cual no es natural. Ası los sistemas de funciones iteradas tienen esteinconveniente. Hubo algunos intentos para resolver este problema pero nofue sino hasta 1989, cuando un estudiante de Michael F. Barnsley, ArnaudJacquin diseno un nuevo metodo de codificacion de imagenes basado en el


concepto fundamental de los SFI, pero haciendo a un lado el enfoque rıgidode los SFI globales.

El nuevo metodo obedecıa a una idea que en principio parece muy sim-ple. En vez de ver a una imagen como una serie de copias transformadas dealgun compacto arbitrario, esta vez la imagen estarıa formada por copiasde pedazos de si misma bajo transformaciones apropiadas. La mejilla denuestra tıa no se parece a un arbol pero es muy probable que su mejilladerecha si sea muy parecida a la izquierda, ası pues esta idea contrarestaen buena manera el problema que los SFI globales tienen. Este metodo seconoce como sistemas de funciones iteradas por partes SFIP.

Figura 3: Existen varias partes de la imagen que se parecen entre si, enparticular las que estan encerradas en las regiones R1 y R2.

La idea general se basa en tomar una particion de la imagen, los ele-mentos de la particion pueden tener formas arbitrarias. En [7], [14], [2] y[9] estan descritos diversos algoritmos para realizar una particion, algunosocupan triangulos o polıgonos de tamano variable, Estos pueden mejorarla calidad de compresion pues facilitan encontrar la auto-semejanza entrelas secciones de la imagen, pero tambien es posible que aumenten los costosde computo. La manera mas sencilla de hacer la segmentacion para unaimplementacion es utilizando cuadrados de tamano uniforme pues es masfacil delimitar los pixeles encerrados por estas regiones para realizar lascomparaciones. Un estudio general del problema anterior se encuentra en[5].

Consideraremos a las imagenes como funciones que estan definidas sobreI2 y que tienen por contradominio a I, donde I := [0, 1]. A cada punto delcuadrado unitario le corresponde un valor que representa su nivel de gris,con esto podrıamos establecer que si a un punto le corresponde el valor 0


en la imagen se representarıa dicho valor como un punto negro, y el valor1 se representarıa en la imagen como un punto blanco.

Sea A ⊂ I2, sabemos que una funcion ϕ : A −→ I es un subconjuntode A × I con la propiedad de que para cada (x, y) ∈ A existe un unicoz ∈ I, tal que ((x, y), z) ∈ ϕ ⊂ A × I y generalmente a z se le denotacomo ϕ(x, y). Por otra parte la grafica de la funcion ϕ es un subconjuntode I3 que consiste de todas las tripletas (x, y, z), tales que x, y ∈ I yz = ϕ(x, y), a la grafica de una funcion ϕ la denotaremos por ∗(ϕ). Comopuede observarse una funcion y su grafica no son lo mismo, pero dada unafuncion, su grafica esta implicitamente definida con esta, y viceversa, siuno conoce la grafica de una funcion puede de inmediato saber quien es lafuncion. Como estaremos trabajando con las funciones antes mencionadasy sus graficas, tomaremos la siguiente notacion en adelante:

F :={ϕ ⊂ A× I | ϕ es una funcion y A ⊆ I2

}

y definiremos a

G := {∗(ϕ) | ϕ ∈ F} .

A cada elemento de F se le puede asociar de manera unica un elementode G , recıprocamente a cada elemento de G se le puede asociar de mane-ra unica un elemento de F , por tanto existe una biyeccion entre ambosconjuntos la funcion que definiremos a continuacion es tal biyeccion.

� : F −→ G

dada por

∀ ϕ ∈ F ; �(ϕ) := ∗(ϕ).

La metrica del supremo la definiremos sobre nuestro conjunto F demanera usual como:

dsup : F × F −→ R+

dada por ∀ ϕ, ψ ∈ F

dsup(ϕ, ψ) := sup {|z1 − z2| | x, y ∈ I, (x, y, z1) ∈ ∗(ϕ) y (x, y, z2) ∈ ∗(ψ)} .

Proposicion 3.1. El espacio metrico (F , dsup) es un espacio metrico com-pleto.

Definicion 3.2. Un sistema de funciones iteradas por partes o SFIP esuna terna

({(F , dsup); g1, . . . , gn)} ,D ,R)


formada por un SFI {(F , dsup); g1, . . . , gn)} , una cubierta finita de I2

D := {D1, . . . , Dk} ,

tal que ∀ i ∈ {1, . . . , k} , Di �= ∅ y una particion de I2

R := {R1, . . . Rn} ,

tales que: Dada ϕ ∈ F y dada i ∈ {1, . . . , n}, existen ψ ∈ F y j ∈{1, . . . , k} tales que

gi(ϕ|Dj) = ψ|Ri

.

Definicion 3.3. Se dice que una familia de funciones

{gi | gi : G −→ G , i = 1, . . . , n}

enlosan4 a I2, si para toda ϕ ∈ F , se tiene que

n⋃i=1

gi(∗(ϕ)) ∈ G .

Definicion 3.4. Sea f : A ⊆ R3 −→ R3 una funcion y f1, f2, f3 : R −→ Rsus funciones coordenadas, es decir

∀ (x, y, z) ∈ R3; f(x, y, z) = (f1(x), f2(y), f3(z)),

entonces diremos que f es una funcion contraıble respecto a su tercer com-ponente, si existe α ∈ [0, 1), tal que ∀ z1, z2 ∈ R,

d(f3(z1), f3(z2)) ≤ α d(z1, z2)

y f1(x), f2(y) son independientes de z1 y de z2, para todo x, y ∈ R. A α sele llama un factor de contraccion respecto a la tercera componente.

Proposicion 3.5. Sean{Di ⊆ I2 | i ∈ {1, . . . , n}

}una cubierta para I2,

{fi : G −→ G | i ∈ {1, . . . , n}}

una familia finita de funciones que enlosan I2, tales que

∀ i ∈ {1, . . . , n} , ∀ ϕ ∈ F ; fi(∗(ϕ|Di)) = fi(x, y, ϕ(x, y)) con x, y ∈ Di

es una funcion contractiva respecto a su tercer componente y αi es un factorde contracccion. Entonces la funcion

F : F −→ F4La palabra enlosar significa cubrir un suelo con losas unidas y ordenadas. Por lo que

preferimos la palabra enlosar en vez de cubrir, para enfatizar que pretendemos fabricaruna cubierta enlosando la imagen con secciones (losas matematicas) de sı misma.


dada por

∀ ϕ ∈ F ; F (ϕ) := �−1

(n⋃

i=1

fi(∗(ϕ))

),

es una contraccion en el espacio metrico (F , dsup) y α := max {α1, . . . , αn}es un factor de contraccion para F .

Demostracion. Recordemos que � es la biyeccion que existe entre F yG .

Queremos probar que existe α ∈ [0, 1) tal que ∀ ϕ, ψ ∈ F , se cumpleque:

dsup(F (ϕ), F (ψ)) ≤ α dsup(ϕ, ψ).

Sean ϕ, ψ ∈ (F , dsup)y sea α = max {α1, . . . , αn}, estimemos la distanciaentre F (ϕ) y F (ψ).

dsup(F (ϕ), F (ψ)) =

= sup {|z − w| | x, y ∈ I, (x, y, z) ∈ ∗(F (ϕ)) y (x, y, w) ∈ ∗(F (ψ))}

= sup{|z − w| | x, y ∈ I, (x, y, z) ∈ ∗(�−1(∪ni=1fi(

∗(ϕ|Di))))

y (x, y, w) ∈ ∗(�−1(∪ni=1fi(

∗(ψ|Di))))}

= sup{|z − w| | x, y ∈ I, (x, y, z) ∈ ∗(�−1(fi(∗(ϕ|Di

))))

y (x, y, w) ∈ ∗(�−1(fi(∗(ψ|Di)))), i ∈ {1, . . . , n}}

= sup{|z − w| | (x, y) ∈ Di, (x, y, z) ∈ fi(∗(ϕ|Di

))

y (x, y, w) ∈ fi(∗(ψ|Di)), i ∈ {1, . . . , n}}

= sup{|fi3(ϕ(x, y))− fi3(ψ(x, y))| | (x, y) ∈ Di, ϕ(x, y) = z

y ψ(x, y) = w, i ∈ {1, . . . , n} .

Por la propiedad de contraccion respecto a la tercera componente,∀ i ∈{1, . . . , n} de fi tenemos.

sup{|fi3(ϕ(x, y))− fi3(ψ(x, y))| | (x, y) ∈ Di, ϕ(x, y) = z y ψ(x, y) = w,ψ(x, y) = w, i ∈ {1, . . . , n}}

≤ sup{αi |ϕ(x, y)− ψ(x, y)| | (x, y) ∈ Di, ϕ(x, y) = z y ψ(x, y) = w,

i ∈ {1, . . . , n}}

≤ sup{α |ϕ(x, y) − ψ(x, y)| | (x, y) ∈ Di, ϕ(x, y) = z y ψ(x, y) = w,i ∈ {1, . . . , n}}


= α sup{|ϕ(x, y)− ψ(x, y)| | (x, y) ∈ I2, ϕ(x, y) = z y ψ(x, y) = w}

= α sup{|z − w| | x, y ∈ I, (x, y, z) ∈ ∗(ϕ) y (x, y, w) ∈ ∗(ψ)}

= α dsup(ϕ, ψ),

por lo tanto

dsup(F (ϕ), F (ψ)) ≤ α dsup(ϕ, ψ).

Ademas α = max {α1, . . . , αn} ∈ [0, 1), ası F es un contraccion en (F , dsup)

tal y como se querıa demostrar. �

4 Implementacion

El modelo matematico de una imagen como una funcion ϕ : I2 −→ Inos permite que la teorıa funcione, sin embargo a la hora de tratar la infor-macion de una imagen con una computadora no necesariamente funcionara,pues hay que hacer la consideracion de que en la computadora la funcionque modela a una imagen debe tener un dominio y rango discretos, de locontrario una imagen representada por una funcion ϕ : I2 −→ I a pesar detener un tamano finito, describirıa a una imagen de resolucion infinita.

A lo que nos referimos en el parrafo anterior es que en una computa-dora se representa una imagen como una coleccion discreta de elementosde pigmento o pixeles, cada pixel toma un valor discreto en una escala degrises o bien tres en una escala con tres canales de color, el numero de bitsusados para almacenar estos valores es lo que se llama la resolucion de laescala. Si usamos 8 bits para almacenar un solo valor por pixel, la imagenpodrıa almacenarse en una computadora como una matriz de tamano n×ndonde cada entrada de la matriz serıa un numero entre 0 y 255, este valorrepresentara un nivel en una escala de grises.

La imagen siguiente es un ejemplo de lo anterior, esta tiene un total de256×256 elementos de pigmento cada uno con un valor entre 0 y 255, paracada pixel se reservan 8 bits que almacenan su valor en binario. En totalpara poder almacenar la imagen en una computadora se ocuparıan, 1 bytepor pixel, lo que serıa 256× 256 = 65 536 bytes o bien 64 kb.


Figura 4: Esta imagen es de 256 por 256 pixeles, la escala de grises es unvalor entero entre 0 y 255, donde 0 es valor de negro y 255 el de blanco.

Ahora supongamos que dividimos esa misma imagen, en una cubiertaD y una particion R ambas con elementos cuadrados de tamano uniforme,los Di ∈ D de tamano 16 × 16 mientras que los Ri ∈ R sean de tamano8× 8.

D R

Figura 5: Segmentacion de la imagen

De esta manera para cada uno de los 16×16 = 256 elementos Ri ∈ R queencierran 16 pixeles se tratara de encontrar un elemento Dj ∈ D de entrelos 8× 8 = 64 y una transformacion tal que esta evaluada en Dj minimicelos valores en la escala de grises con respecto al Ri en cuestion, notemos quede entrada tanto los elementos de D como los de R son cuadrados con laparticularidad de que lo primeros son del doble del tamano de los segundos,la tranformacion geometrica tendra que ser por fuerza un escalamiento deun elemento de D a la mitad de su tamano, y una traslacion a la posiciondel elemento de R en turno, pero hay ocho posibles formas de mapear un


cuadrado en otro, cuatro rotaciones y cuatro reflexiones: vertical, horizontaly respecto a sus dos diagonales.

Para cada Ri, tal que i = 1, . . . , 256 debemos tomar

mink=1,...,64

{min

j=1,...,8{d(fki(ϕ(Dk)), ϕ(Ri))}

}.

Lo anterior es costoso hablando en terminos de computo ya que se re-quiere hacer para este caso un total de 8×64 = 512 comparaciones por cadaelemento de R ası que en resumen son 512×256 = 131 072 comparaciones.

El proceso de codificacion no es muy complicado y se puede implementarcon el siguiente algoritmo.

1. - Leer la imagen a ser codificada (traducirla en la matriz antes co-mentada).

2. - Segmentar la imagen en una cubierta D y una particion R.

3. - Para un R0 ∈ R dado, compararlo con cada una de las ocho posiblesformas en que se puede mapear cada uno de los elementos de D enR0. Obtener y almacenar el mejor D0 ∈ D y la mejor transformacionque aproximan a R0.

4. - Repetir el paso anterior para cada uno de los elementos de nuestraparticion R.

El proceso de decodificacion no es tan tardado comparado con el decodificado, como sabemos de la teorıa desarrollada tenemos que iterar lafuncion evaluada en una imagen cualquiera y obtener el punto fijo, el cuales el lımite de esta sucesion de iteraciones. Veremos en los ejemplos que lasucesion se aproxima bastante rapido al punto fijo.

El algoritmo de decodificacion es el siguiente:

1. - Leer los coeficientes de las funciones ası como los Di.

2. - Crear una imagen cualquiera del mismo tamano de la imagen origi-nal.

3. - Tomar la cubierta R como en el paso de codificacion, y a cada Ri

aplicarle la transformacion correspondiente.

4. - Hacer la imagen que resulta en el paso anterior la nueva entradapara el algoritmo hasta la iteracion deseada.


Segun el numero de iteraciones la calidad de la imagen reconstruida vaevolucionando para mejorar como puede observarse en la figura 6.

(a)

(b) (c)

Figura 6: La figura 4 fue comprimida y en el proceso de reconstruccion resultanla imagen (a) haciendo una iteracion, la imagen (b) al hacer cuatro iteraciones y(c) realizando ocho iteraciones.

Mostraremos mas ejemplos como las imagenes 6 y 7 que fueron tomadasde [11] y las imagenes 8 y 9 de [12]. Para poder entender los resultadosque se obtienen con los algoritmos anteriores al procesar estas imagenesnecesitamos conocer lo que es el PSNR (Peek Signal to Noise Ratio), quees una de las formas mas conocidas de medir en decibeles la calidad deuna imagen reconstruida con respecto a una imagen original despues de unproceso de compresion. Tambien se requiere conocer el error cuadraticomedio o MSE (Mean Square root Error). Supongamos que tenemos dosimagenes distintas I y F de tamano n×m, entonces las cantidades anterioresse definen de la siguiente manera:

MSE :=1

nm

n−1∑i=0

m−1∑j=0

||I(i, j)− F (i, j)||2,


PSNR := 10 · log10

(MAX2

F

MSE

)= 20 · log10

(MAXF

MSE

).

Donde MAXF es el valor maximo que puede tomar un pixel en la imagenF .

Imagen original Imagen reconstruida

Figura 7: La imagen original es de 512×512 pixeles, la imagen reconstruida tieneun PSNR = 30.00 dB y una razon de compresion de 70.29. Tiempo de codificado1.7 segundos.


Figura 8: La imagen original es de 512×512 pixeles, la imagen reconstruida tieneun PSNR= 29.91 dB y una razon de compresion de 70.30. Tiempo de codificado1.1 segundos.



Figura 9: La imagen original es de 512×512 pixeles, la imagen reconstruida tieneun PSNR = 33.19 dB y una razon de compresion de 7.93. Tiempo de codificado2.12 segundos.


Figura 10: La imagen original es de 256 × 256 pixeles, la imagen reconstruidatiene un PSNR = 32.75 dB y una razon de compresion de 5.86. Tiempo decodificado 0.7 segundos.

5 Conclusiones

La tecnica de compresion de imagenes digitales mediante conjuntos fractalesgenerados por sistemas de funciones iteradas por partes es una ingeniosaaplicacion de las matematicas a la informatica. Este trabajo solo introduce


al lector en la implementacion computacional de dicha tecnica. Cabe men-cionar que existen estudios en [5] en donde se trata el problema de mejorarel algoritmo en sus distintos pasos para lograr un desempeno mas eficiente.Ademas de la compresion de imagenes, existen tecnicas de super-resoluciona diferentes escalas de una imagen, estas tecnicas tambien se ayudan de laauto-semejanza o redundancia de la imagen [6].

Sergio Mabel Juarez VazquezCIDETEC,[email protected]

Flor de Marıa Correa RomeroDepartamento de Matematicas,Escuela Superior de Fısica y [email protected]

Referencias

[1] Barnsley M., Fractals Everywhere, Academic Press, New York 1988.

[2] Davione F.; Svensson J.; Chassery J., A mixed triangular and quadrilateralpartition for fractal image compression, Proceedings of International Con-ference on Image Processing, Washington, DC (1995), No. 3, 284–287.

[3] Ebrahimi M.; Vrscay E., Self-similarity in imaging, 20 years after ”Frac-tals Everywhere”, Proceedings of International Workshop on Local and Non-Local Approximation in Image Processing, Lausanne (2008), 165–172.

[4] Falconer K., Fractal Geometry: Mathematical Foundations and Applica-tions, John Wiley and Sons, 2003.

[5] Fisher Y., Fractal Image Compression: Theory and Application, Springer-Verlag, 1995.

[6] Glashner D.; Bagon S.; Irani M., Super-resolution from a single image, Pro-ceedings of 12th International Conference on Computer Vision, 2009, 349–356.

[7] Harstenstein H.; Saupe D., Cost-based region growing for fractal image com-pression, Proceedings of IX European Signal Processing Conference, Rhodes1998, 2313–2316.

[8] Hutchinson J.E., Fractals and self-similarity, Indiana Univ. Math. J. No. 30(1981), 713–747.

[9] Kramm M., Image cluster compression using partitioned iterated functionsystems and efficient inter-image similarity features, Third InternationalIEEE Conference on Signal-Image Technologies and Internet-Based System,Shanghai 2007, No. 1, 989–996.

[10] Mandelbrot B., The Fractal Geometry of Nature, W.H. Freeman, 1977.

[11] Ochotta T. y Saupe D., Edge-based partition coding for fractal image com-pression, Arab. J. Sci. Eng., Special Issue on Fractal and Wavelet Methods(2004).


[12] Perez J., Codificacion fractal de imagenes, Universidad de Alicante, Espana,Tech. Rep., 1997. Una version del documento esta disponible en la direccionhttp://www.dlsi.ua.es/∼japerez/pub/pdf/mastertesi1998.pdf

[13] Rudin W., Real and Complex Analysis, MacGraw-Hill 1986.

[14] Saupe D.; Ruhl M., Evolutionary fractal image compression, IEEE Interna-tional Conference on Image Processing, Lausanne 1996, 129–132.


Uniqueness of the Solution of the Yule-Walker

Equations: A Vector Space Approach ∗

Ana Paula Isais-Torres 1 Rolando Cavazos-Cadena 2

Abstract

This work concerns the Yule-Walker system of linear equationsarising in the study of autoregressive processes. Given a com-plex polynomial ϕ(z) satisfying ϕ(0) = 1, elementary vector spaceideas are used to derive an explicit formula for the determinantof the matrix M(ϕ) of the Yule-Walker system of equations cor-responding to ϕ. The main conclusion renders the following non-singularity criterion: The matrix M(ϕ) is invertible if and only ifthe product of two roots of ϕ is always different form 1, a propertythat yields that the Yule-Walker system associated with a causalpolynomial has a unique solution. The way in which this result isimplicitly used in the time series literature is briefly discussed.

2000 Mathematics Subject Classification: 37M10.Keywords and phrases: autoregressive processes, Yule-Walker equations,causal polynomials, unique solution, determination of an autocovariancefunction.

1 Introduction

This note concerns a basic question involving the Yule-Walker equationsin time series analysis. To describe the problem we are interested in,

∗This work is part of the M. Sc. Thesis in Statistics of the first author under thesupervision of the second author. The degree was granted in february 2012 by theUniversidad Autonoma Agraria Antonio Narro.

1Author partially supported by a scholarship of CONACyT.2Author supported in part by the PSF organization under grant 105657.

29


Uniqueness of the Solution of the Yule-Walker

Equations: A Vector Space Approach ∗

Ana Paula Isais-Torres 1 Rolando Cavazos-Cadena 2

Abstract

This work concerns the Yule-Walker system of linear equationsarising in the study of autoregressive processes. Given a com-plex polynomial ϕ(z) satisfying ϕ(0) = 1, elementary vector spaceideas are used to derive an explicit formula for the determinantof the matrix M(ϕ) of the Yule-Walker system of equations cor-responding to ϕ. The main conclusion renders the following non-singularity criterion: The matrix M(ϕ) is invertible if and only ifthe product of two roots of ϕ is always different form 1, a propertythat yields that the Yule-Walker system associated with a causalpolynomial has a unique solution. The way in which this result isimplicitly used in the time series literature is briefly discussed.

2000 Mathematics Subject Classification: 37M10.Keywords and phrases: autoregressive processes, Yule-Walker equations,causal polynomials, unique solution, determination of an autocovariancefunction.

1 Introduction

This note concerns a basic question involving the Yule-Walker equationsin time series analysis. To describe the problem we are interested in,

∗This work is part of the M. Sc. Thesis in Statistics of the first author under thesupervision of the second author. The degree was granted in february 2012 by theUniversidad Autonoma Agraria Antonio Narro.

1Author partially supported by a scholarship of CONACyT.2Author supported in part by the PSF organization under grant 105657.

29

30 A. P. Isais-Torres and R. Cavazos-Cadena

let {Xt} be a zero-mean (second order) stationary process which is sup-posed to be real-valued and autoregressive of order p (AR(p)), that is,{Xt} satisfies a difference equation of the form

Xt + ϕ1Xt−1 + · · ·+ ϕpXt−p = Zt,(1)

where the Zt’s are zero-mean uncorrelated random variables with com-mon variance σ2 > 0; see, for instance, Anderson (1971) pp. 166–176,or Box and Jenkins (1976), pp. 53–65. It is known that a process{Xt} satisfying (1) exists if and only if the autoregressive polynomialϕ(z) := 1 + ϕ1z + · · · + ϕpz

p is such that ϕ(z) �= 0 for all complex zwith |z| = 1, and in this case, there is not loss of generality in assumingthat the polynomial ϕ(z) is causal, i.e., ϕ(z) �= 0 for all z with |z| ≤ 1,a condition that is supposed to hold in the following discussion; for de-tails, see Remarks 3 and 5 in Brockwell and Davis (1987), pp. 86–88.Now, consider the problem of determining the autocovariance functionγ(·) of {Xt}, which is given by γ(h) := Cov (Xt+h, Xt), h = 0,±1, . . ..Multiplying both sides of (1) byXt−i and taking the expectation in bothsides of the resulting equality, it follows that

∑pk=0 γ(|i− k|)ϕk = 0 for

i > 0, and∑p

k=0 γ(k)ϕk = σ2 if i = 0, where ϕ0 = ϕ(0) = 1. Thefollowing two-step method, which is described in Brockwell and Davis(1987), p. 97, is a computationally convenient tool to determine γ(·).Step 1. Find γ(0), γ(1), . . . , γ(p) by solving

p∑k=0

γ(k)ϕk = σ2

p∑k=0

γ(|i− k|)ϕk = 0,(2)

which is the Yule-Walker (Y-W) system of equations associated to ϕ(·).Step 2. Using that γ(i) = −

∑pk=1 γ(i − k)ϕk for i > p, determine

γ(p+ 1), γ(p+ 2), . . . in a recursive way.

In order to have that the above method relies on firm grounds, it isnecessary to show that (2) has a unique solution, a fact that can beeasily verified when the degree of ϕ(z) is small, say p = 1 or p = 2; seeSection 3 below. For a polynomial of arbitrary degree, Achilles (1987)used an argument based on matrix theory to obtain a formula for thedeterminant of matrix M(ϕ) of the above Y-W linear system associ-ated to a polynomial ϕ, and a more compact and advanced approach

Unique Solution of the Yule-Walker System 31

was used in Lutkepohl and Maschke (1988). On the other hand, it isinteresting to observe that the Yule–Walker equations are used (i) tofind moment estimators of the polynomial ϕ (see, for instance, Section8.2 in Brockwell and Davis, 1991) and (ii) To compute recursively thecovariance function of a general ARMA(p, q) process (Brockewll andDavis, 1991, Cavazos-Cadena, 1994). Also, recently an efficient way toimplement the innovations algorithm to construct best linear predictorsvia the Durbin-Levinson algorithm (using the non-singularity of M(ϕ)for a causal polynomial ϕ) was obtained in Martınez–Martınez (2010).

The main objective of this work is to present an elementary deriva-tion of the determinant of the matrix M(ϕ) using simple vector spacesideas. The result in this direction, presented in Theorem 3.1 below,yields the following criterion: The Y-W system associated with thepolynomial ϕ(·) has a unique solution if and only if

rirj �= 1, i, j = 1, 2, . . . , p,(3)

where r1, . . . , rp are the roots of ϕ(z). Notice that when ϕ(z) is a causalpolynomial all of its roots ri lie outside the unit disk, and in this case(3) is clearly satisfied.

The proof of Theorem 3.1 is based on an induction argument usingstandard ideas on vector spaces, and the exposition has been organizedas follows: In Section 2 the basic (infinite–dimensional) vector space isintroduced, and the main result on the determinant of the matrix M(ϕ)is stated in Section 3. Next, in Section 4 some necessary preliminariesto prove Theorem 3.1 are presented, and the proof of the main result isgiven in Section 5. Finally, the presentation concludes with some briefcomments in Section 6.

2 Auxiliary Vector Space and Basic Notation

Throughout the remainder Z and IN stand for the sets of all integersand all nonnegative integers, respectively, and C denotes the set of allcomplex numbers. The complex vector space L consists of all vectorsv : IN → C with the property that v(k) = 0 for all k large enough, andL is endowed with the usual addition and scalar multiplication. Theshift operator s:L → L is defined as follows: For v ∈ L

s(v)(0) := 0, and s(v)(k) := v(k − 1), k = 1, 2, . . . ;(4)


in addition, the n-fold composition of the shift operator with itself isdenoted by sn, so that

s0(v) = v, and sn(v) = sn−1(s(v)).(5)

On the other hand, rows and columns of a squared matrix M arenumbered starting from zero, and Det M denotes the determinant ofM . For vectors v0,v1, . . . ,vn ∈ L, the corresponding squared matrixMn+1(v0,v1, . . . ,vn) of order (n + 1) is defined by

Mn+1(v0,v1, . . . ,vn) := vi(j), i, j = 0, 1, . . . , n,(6)

whereas span{v0, . . . ,vn} stands for the vector space generated by thevectors v0, . . . ,vn; the corresponding dimension is denoted by

dim span{v0, . . . ,vn}.

To conclude, with a given a polynomial ϕ(z) = ϕ0+ϕ1z+ · · ·+ϕpzp of

degree p, the vectors −→ϕ , ←−ϕ ∈ L are defined as follows:

For k = 0, 1, . . . , p −→ϕ (k): = ϕk and ←−ϕ(k): = ϕp−k,

and, for k > p, −→ϕ (k) = ←−ϕ(k) = 0.(7)

Finally, the following notational convention concerning the coefficientsof the polynomial ϕ(z) will be used:

ϕk := 0 for k < 0 or k > p.(8)

3 Main Result

Let ϕ(z) be a polynomial of degree p. The objective of this section isto state a formula for the determinant of the matrix corresponding tothe Y-W system (2). To begin with, notice that for i > 0,

p∑k=0

γ(|i− k|)ϕk =i∑

k=0

γ(i− k)ϕk +

p∑k=i+1

γ(k − i)ϕk

=i∑

j=0

γ(j)ϕi−j +

p−i∑j=1

γ(j)ϕi+j

and using convention (8) it follows that

p∑k=0

γ(|i− k|)ϕk = γ(0)ϕi +

p∑j=1

γ(j)[ϕi−j + ϕi+j ].


Thus, the Y-W system (2) can be equivalently written as

p∑k=0

γ(k)ϕk = σ2

γ(0)ϕi +

p∑j=1

γ(j)[ϕi−j + ϕi+j ] = 0, i = 1, 2, . . . , p.(9)

The (squared) matrix of this system will be denoted by M(ϕ). Clearly,M(ϕ) is of order (p+ 1) and is given as follows: For i = 0, 1, 2, . . . , p,

M(ϕ)i 0 := ϕi, and M(ϕ)i j := ϕi−j + ϕi+j , j = 1, 2, . . . , p;(10)

for instance, M(ϕ)0 0 = ϕ0 and, for j > 0, M(ϕ)0 j = ϕ0−j +ϕ0+j = ϕj ,in accordance with the first equation in (9). The next theorem containsa formula for the determinant of M(ϕ) and, as a by-product, a criterionfor the non-singularity of M(ϕ) is obtained.

Theorem 3.1. Let ϕ(z) = 1+ϕ1z+ · · ·+ϕpzp be a complex polynomial

of degree p. If the roots of ϕ(·) are r1, . . . , rp and M(ϕ) is as in (10),then assertions (i) and (ii) below occur.

(i) The determinant of M(ϕ) is given by

Det M(ϕ) =∏

1≤i<j≤p

[1− (rirj)−1]

p∏i=1

(1− r−2i )(11)

where, by (the usual) convention, for p = 1 the first product in the abovedisplay is 1.

Consequently,

(ii) M(ϕ) is non-singular if and only if rirj �= 1 for all i, j = 1, . . . , p.

This result will be established in Section 5; by the moment, it isconvenient to note that (ii) follows immediately from part (i). On theother hand, (11) is easily verified for small values of p. For instance, forp = 1, the polynomial ϕ(z) is given by ϕ(z) = 1 + ϕ1z and (10) yieldsthat

M(ϕ) =

[1 ϕ1

ϕ1 1

]

so that Det M(ϕ) = 1− ϕ21 , and this yields (11) with p = 1, since ϕ(·)

has the unique root r1 = −1/ϕ1. For p = 2 factorize ϕ(z) as

ϕ(z) = (1 + a1z)(1 + a2z),


where the roots of ϕ(z) are ri = −1/ai, i = 1, 2. It follows that

ϕ(z) = 1 + (a1 + a2)z + a1a2z2

and M(ϕ) is given by

M(ϕ) =

1 a1 + a2 a1a2a1 + a2 a1a2 + 1 0a1a2 a1 + a2 1

Then, expandingDetM(ϕ) by the third column the following expressionis obtained:

Det M(ϕ) = a1a2[(a1 + a2)− a1a2(1 + a1a2)]

+(1 + a1a2)− (a1 + a2)2

= −(a1 + a2)2(1− a1a2) + (1 + a1a2)[1− (a1a2)

2]

= (1− a1a2)[−(a1 + a2)2 + (1 + a1a2)

2]

= (1− a1a2)(1− a21)(1− a22)

and replacing ai by −1/ri the formula in (11) is obtained for the casep = 2. The proof of (11) in the general case is by induction and ispresented in Section 5.

4 Preliminaries

This section contains the technical tools that will be used to establishTheorem 3.1. The starting point is the idea in the following definition.

Definition 4.1. . Let ϕ(z) = 1 + ϕ1z + · · · + ϕpzp be a polynomial of

degree p. The sequence Vϕ = {V ϕt | t ∈ Z} ⊂ L is defined as follows:

(i) For 0 ≤ n < p,

V ϕn (0) := ϕn, and V ϕ

n (k) := ϕn−k + ϕn+k, k = 1, 2, . . . .

(ii) For n ∈ IN

V ϕ−n: = sn(−→ϕ ) and V ϕ

n+p: = sn(←−ϕ);

see (4)–(7) for notation.


A glance to the above definition and (4)–(8) shows that the sequenceVϕ is related to the matrix M(ϕ) through the following equality:

M(ϕ) = Mp+1(Vϕ0 , V ϕ

1 , . . . , V ϕp ).(12)

The following is the key technical result of this section.

Theorem 4.2. Suppose that ϕ(z) = 1 + ϕ1z + · · ·+ ϕpzp has degree p

and satisfies ϕ(b) = ϕ(1/b) = 0 for some b ∈ C \ {0}. In this case thefollowing assertions (i)–(iii) occur.

(i) dim span{V ϕ−1, V

ϕ0 , . . . , V ϕ

p+1} ≤ p+ 1.

(ii) For each a ∈ C,

Det Mp+2(Vϕ0 + aV ϕ

−1, Vϕ1 + aV ϕ

0 , . . . , V ϕp+1 + aV ϕ

p ) = 0.

(iii) For all a ∈ C, Det M [(1 + az)ϕ(z)] = 0.

The proof of Theorem 4.1 has been divided into several pieces pre-sented in the Lemmas 4.1–4.4 below, which involve the notion intro-duced below.

Definition 4.3. Let V = {Vt|t ∈ Z} be a sequence in L.

(i) V has property D(p) if for all n ∈ IN

dim span{Vt| − n ≤ t ≤ p+ n} ≤ p+ n.

(ii) Given a ∈ C, the sequence TaV = {TaVt | t ∈ Z} is defined by

[TaV]t ≡ TaVt := Vt + aVt−1, t ∈ Z.

The starting point of the journey to the proof of Theorem 4.2 is thefollowing.

Lemma 4.4. Let ϕ(z) = 1+ϕ1z+ · · ·+ϕpzp be a polynomial of degree

p and a ∈ C \ {0}. If θ(z) = (1 + az)ϕ(z), then Vθ = TaVϕ

Proof. Let n ∈ {1, 2, . . . , p} be arbitrary. From Definition 4.1 it followsthat

V θn (0) = θn = ϕn + aϕn−1 = V ϕ

n (0) + aV ϕn−1(0) = TaV

ϕn (0),


and for k = 1, 2, . . .,

V θn (k) = θn+k + θn−k

= (ϕn+k + aϕn−k−1) + (ϕn−k + aϕn−k−1)

= (ϕn+k + ϕn−k) + a[ϕn−1−k + ϕn−1+k]

= V ϕn (k) + aV ϕ

n−1(k)

= TaVnϕ(k)

These two last displays show that, for 1 ≤ n ≤ p, V θn = TaV

ϕn , and

to complete the proof this equality should be verified for n < 0 andn > p. To achieve this goal, first notice that

−→θ = −→ϕ + as(−→ϕ ) and←−

θ = s(←−ϕ) + a←−ϕ, a fact that can be obtained from (4) and (7). Then,for n ≥ 0,

V θ−n = sn(

−→θ )

= sn[−→ϕ + as(−→ϕ )]

= sn(−→ϕ ) + asn+1(−→ϕ )

= V ϕ−n + aV ϕ

−n−1

= TaVϕ−n

Similarly,

V θn+p+1 = sn(

←−θ )

= sn[s(←−ϕ) + a←−ϕ]= sn+1(←−ϕ) + asn(←−ϕ)= V ϕ

n+1+p + aV ϕn+p

= TaVϕn+1+p.

Thus, it has been established that V θ−n = TaV

ϕ−n and V θ

n+1+p =TaV

ϕn+1+p for all n ∈ IN; as already noted, this completes the proof. �

The following lemma studies the relation of property D(k) with thetransformation Ta; see Definition 4.3.

Lemma 4.5. Let a ∈ C be arbitrary and suppose that V = {Vt | t ∈Z} ⊂ L has property D(k). Then TaV has property D(k + 1)

Proof. Notice that TaVt ∈ span{Vt, Vt−1} and then, for arbitrary r ∈ IN,span{TaVt| − r ≤ t ≤ k+1+ r} ⊂ span{Vt| − (r+1) ≤ t ≤ k+(r+1)}.


Now observe that, since the sequence V has the property D(k), thespace in the right-hand side of the previous equality has dimension lessthat or equal to k + r + 1 and, consequently,

dim span{TaVt | − r ≤ t ≤ (k + 1) + r} ≤ (k + 1) + r,

a relation that shows that TaV has property D(k + 1), since r wasarbitrary. �

The next two lemmas relate property D(k) with sequences of theform Vϕ; see Definition 4.1.

Lemma 4.6. Let ϕ(z) be a polynomial of degree p ≥ 1 and suppose thatϕ(1) = 0 or ϕ(−1) = 0. In this case Vϕ has property D(p).

Proof. First suppose that p = 1. When ϕ(1) = 0 the polynomial ϕ(z)is given by ϕ(z) = ϕ(0)(1− z) and, using (8), it follows that −→ϕ = −←−ϕ,which yields that, for n ≥ 0, V ϕ

−n = sn(−→ϕ ) = −sn(←−ϕ) = −V ϕn+1. If

ϕ(−1) = 0 it follows that ϕ(z) = ϕ(0)(1 + z) and then −→ϕ = ←−ϕ, andthen V ϕ

−n = V ϕn+1, n ∈ IN. Therefore, in either case, for every r ∈ IN,

span{V ϕt | − r ≤ t ≤ 1 + r} = span{V ϕ

t | 1 ≤ t ≤ 1 + r}

and since the space in the right-hand side has r+1 generators, it followsthat dim span{V ϕ

t | −r ≤ t ≤ 1+r} ≤ r+1, i.e, Vϕ has property D(1);see Definition 4.1. The proof will be now completed by induction in p.Suppose that the result holds for p = k, and let ϕ be a polynomial ofdegree k + 1 vanishing at 1 or −1. In this case it is clearly possible tofactorize ϕ(z) as ϕ(z) = (1 + az)θ(z), where a ∈ C and θ(z) has degreek and satisfies θ(1) = 0 or θ(−1) = 0. By the induction hypothesis,Vθ has property D(k) and, using Lemmas 4.4 and 4.5, it follows thatVϕ = TaV

θ has the property D(k + 1). �

The following is the final step before the proof of Theorem 4.1.

Lemma 4.7. Let ϕ(z) be a polynomial of degree p ≥ 2 such that ϕ(b) =ϕ(1/b) = 0 for some b ∈ C \{0, 1,−1}. Then, Vϕ has the property D(p)

Proof. The argument is along the same lines as in the proof of Lemma4.6. First suppose that p = 2. In this case

ϕ(z) = ϕ(0)(1− bz)(1− z/b) = ϕ(0)[1− (b+ b−1)z + z2],


and using (7) it follows that −→ϕ = ←−ϕ; by Definition 4.1 (i) this yieldsthat V ϕ

−n = V ϕ1+n for every n ≥ 0, and then, for each r ∈ IN,

span{V ϕt | − r ≤ t ≤ 2 + r} = span{V ϕ

t | 1 ≤ t ≤ 2 + r},

and since the vector space in the right-hand side has r + 2 generatorsthis yields that

dim span{V ϕt | − r ≤ t ≤ 2 + r} ≤ r + 2,

that is, Vϕ has the property D(2). The result for arbitrary p is obtainedby an induction argument similar to that used in the proof of Lemma4.6. �

The previous lemmas are used below to establish the main result ofthis section.

Proof of Theorem 4.2. Let ϕ(z) = 1+ϕ1z+ · · ·+ϕpzp be a polynomial

of degree p with ϕ(b) = ϕ(1/b) = 0 for some b ∈ C \ {0}

(i) When b = 1 or b = −1 Lemma 4.6 yields that Vϕ has property D(p),and Lemma 4.7 implies that the same conclusion holds when b �= 1,−1.Then, by Definition 4.3(i), it follows that

dim span{V ϕ−1, V

ϕ0 , . . . , V ϕ

p , V ϕp+1} ≤ p+ 1.

(ii) Notice that

span{V ϕr + aV ϕ

r−1 | r = 0, 1, . . . , p+ 1} ⊂ span{V ϕt | − 1 ≤ t ≤ p+ 1},

and then dim span{V ϕr + aV ϕ

r−1 | r = 0, 1, . . . , p + 1} ≤ p + 1, by part(i). It follows that the p+ 2 vectors V ϕ

r + aV ϕr−1, r = 0, 1, 2, . . . , (p+ 1)

are linearly dependent in L, a fact that implies the linear dependenceof the rows of Mp+2(V

ϕ0 + aV ϕ

−1, . . . , Vϕp+1 + aV ϕ

p ); as a consequence,

Det Mp+2(Vϕ0 + aV ϕ

−1, . . . , Vϕp+1 + aV ϕ

p ) = 0;

see, for instance, Chapter 5 of Hoffman and Kunze (1971).

(iii) Set ψ(z) := (1 + az)ϕ(z). In this case ψ(z) has degree p + 1, andusing (12) with p + 1 and ψ instead of p and ϕ, respectively, it followsthat that

M(ψ) = Mp+2(Vψ0 , V ψ

1 , . . . , V ψp+1)

= Mp+2(Vϕ0 + aV ϕ

−1, . . . , Vϕp+1 + aV ϕ

p )


where Lemma 4.4 was used to set the second equality; from this point,the previous part yields that Det M(ψ) = 0. �

This section concludes with a simple a simple fact that will be usefulin the proof of Theorem 3.1.

Lemma 4.8. Let ϕ(z) be a polynomial of degree p with ϕ(0) = 1. Inthis case

Det M(ϕ) = Det Mp+2(Vϕ0 , V ϕ

1 , . . . , V ϕp+1).

Proof. By convenience set L := Mp+2(Vϕ0 , V ϕ

1 , . . . , V ϕp+1) and observe

the following facts (a) and (b).

(a) A glance to (6) and Definition 4.1(i) shows that the submatrix ob-tained by deleting the last row and the last column of L is

Mp+1(Vϕ0 , V ϕ

1 , . . . , V ϕp ).

Next, the components in the last column of L will be evaluated. Firstrecall Definition 4.2(i) and notice that L0 p+1 = V ϕ

0 (p+1) = s0(−→ϕ )(p+1) = −→ϕ (p+ 1) = 0; see (5) and (7). On the other hand, for 1 ≤ n < p,Ln p+1 = ϕn+p+1+ϕn−p−1 = 0, where convention 8) was used to set thelast equality. Finally, Lp p+1 = V ϕ

p (p+1) = s0(←−ϕ)(p+1) = ←−ϕ(p+1) = 0,and Lp+1 p+1 = s(←−ϕ)(p + 1) = ←−ϕ(p) = ϕ0 = ϕ(0) = 1. Summarizing:

(b) The last column of L consists entirely of zeros except by the element

in the last row, which is one. To conclude, expand Det L across the last

column. In this case the facts (a) and (b) above together imply thatDet L = Mp+1(V

ϕ0 , V ϕ

1 , . . . , V ϕp ) = Det M(ϕ), where the last equality

follows from (12). �

5 Proof of Theorem 3.1

The preliminary results in the previous section will be used to establishthe main result of this note.

Proof of Theorem 3.1. As already mentioned it is sufficient to provepart (i). Let ϕ(z) = 1 + ϕ1z + · · · + ϕpz

p be a polynomial of degree pand factorize ϕ as

ϕ(z) =

p∏i=1

(1 + aiz),


where the roots of ϕ(z) are −1/ai, i = 1, 2, . . . , p. With this notation(11) is equivalent to

Det M

[p∏

i=1

(1 + aiz)

]=

∏l≤i<j≤p

[1− aiaj ]

p∏i=1

(1− a2i ),

an equality that was verified in Section 3 for p = 1 and p = 2. Theproof of (5) will be completed by induction. Suppose that (5) holds forp = n ≥ 2 and let a1, a2, . . . , an+1 be no-null complex numbers. Define

ψ(z) :=

n∏i=1

(1 + aiz)

and, for each c ∈ C, set

F (c) := Det Mn+2(Vψ0 + cV ψ

−1, . . . , Vψn+1 + cV ψ

n ).

Combining Lemma 4.4 and (12) it follows that

(a) F (c) = Det M [(1 + cz)ψ(z)]; in particular,

F (an+1) = Det M

[n+1∏i=1

(1 + aiz)

].

Using the multilinearity of the determinant function, (5) yields that

(b) F (c) is a polynomial in c with degree ≤ n + 2; see, for instance,Hoffmann and Kunze (1971) Chapter 5.

Next, the roots of the polynomial F (c) will be determined. Firstobserve that

(c) F (1) = F (−1) = 0.

To verify this last assertion set ψ∗(z) := (1 + z)∏p

i=2(1 + aiz) andnotice that ψ∗(−1) = 0, and (1+z)ψ(z) = (1+a1z)ψ

∗(z); see (5). Thenthe above property (a) yields that

F (1) = Det M [(1 + z)ψ(z)] = Det M [(1 + a1z)ψ∗(z)] = 0,

where the last equality is due to Theorem 4.2(iii) with ψ∗(z) and −1instead of ϕ(z) and b, respectively. Similarly, it can be shown thatF (−1) = 0.

(d) F (1/ai) = 0 for i = 1, 2, . . . , n. To show this, let k, i ∈ {1, 2, . . . , n}


be fixed integers with k �= i and define

ψ(z): = (1 + z/ai)

(k)∏(1 + ajz),

where

(k)∏indicates the product over all the integers j between 1 and n

satisfying j �= k; notice that

(1 + z/ai)ψ(z) = (1 + akz)ψ(z).(13)

From (5) it is clear that ψ(−ai) = 0 and, since k �= i, the polynomialψ(z) contains the factor (1 + aiz), and then ψ(−1/ai) = 0. Therefore,from (a) and (13) it follows that F (1/ai) = Det M [(1 + z/ai)ψ(z)] =Det M [(1 + akz)ψ(z)], and an application of Theorem 4.1(iii) with ψand −ai instead of ϕ and b, respectively, yields that F (1/ai) = 0.

To continue, suppose by the moment that a1, a2, . . . , an are differentnumbers in C \ {0, 1,−1}. In this case, the above facts (c) and (d)together show that the polynomial F (c) has n+2 different roots, namely,1,−1, and 1/ai , i = 1, 2, . . . , n. Combining this fact with (b) it followsthat F (·) has degree n+ 2 and it can be factorized as

F (c) = F (0)(1− c)(1 + c)n∏

i=1

(1− aic).

Setting c = an+1 and using (a) it follows that

Det M

[n+1∏i=1

(1 + aiz)

]= F (0)(1− a2n+1)

n∏i=1

(1− aian+1).

Next, using (5) it follows that F (0) = Det Mn+2(Vψ0 , V ψ

1 , . . . , V ψn+1),

and then F (0) = M(ψ), by Lemma 4.8 applied to ψ, which has degreen, and then the induction hypothesis yields that

F (0) =n∏

i=1

(1− a2i )∏

1≤i<j≤n

(1− aiaj),

and combining this equality with (5) it follows that

Det M

[n+1∏i=1

(1 + aiz)

]=

n+1∏i=1

(1− a2i )∏

1≤i<j≤n+1

(1− aiaj),(14)


which is (5) with p = n+ 1. Although (14) has been established underthe assumption that a1, . . . , an+1 are different numbers in C \{0, 1,−1},the equality holds for arbitrary ai, . . . , an+1 ∈ C \ {0}, since both sidesof (14) are continuous functions of the ais. In short, assuming that (5)holds for p = n it has been shown that it is also valid for p = n + 1completing the induction argument.

6 Concluding Remarks

Given a polynomial ϕ(z) with ϕ(0) = 1, a necessary and sufficient con-dition was established so that the corresponding Yule-Walker systemhas a unique solution, and it has been shown that such a condition issatisfied by a causal polynomial. Besides to provide a rigorous basis forthe two step method described in Section 1, there are other parts in thetheory of time series where it is important to know that a Yule-Walkersystem has a unique solution. For instance, consider the following re-sult: Given p >0 and an autocovariance function γ(·) with γ(h) → 0 ash → ∞ there exists an AR(p) process {Yt} whose autocovariance func-tion γY (·) coincides with γ(·) at lags h = 0, 1, . . . , p. A proof of this factcan be found in Brockwell and Davis (1987), pp. 232-233, and here wejust mention that there is a passage where, for a certain causal polyno-mial ϕ(·), it is shown that both {γ(h)|0 ≤ h ≤ p} and {γ(h) | 0 ≤ h ≤ p}satisfy (2), and then it is immediately concluded that γ(h) = γY (h) for0 ≤ h ≤ p. Thus, it is implicitly assumed that the Yule-Walker systemof a causal polynomial has a unique solution, a fact that, by Theorem3.1, is indeed true.

Acknowledgement

The authors are deeply grateful to the reviewers for their suggestionsto improve the paper.

Ana Paula Isais-TorresSubdireccion de PostgradoUniversidad Autonoma AgrariaAntonio NarroCalzada Antonio Narro 1923,BuenavistaSaltillo Coah. 25315, Mexico

Rolando Cavazos-CadenaDepartamento de Estadıstica yCalculoUniversidad Autonoma AgrariaAntonio NarroCalzada Antonio Narro 1923,BuenavistaSaltillo Coah. 25315, Mexico


References

[1] Achilles M., Zur Lsung der Yule-Walker-Gleichungen, Metrika34 (1987), 237–251.

[2] Anderson T. W., The Statistical Analysis of Time Series, Wiley,New York, (1987).

[3] Box G. E. P. and G. M. Jenkins, Time Series Analysis, Forecastingand Control , Holden-Day, San Francisco CA, (1976).

[4] Brockwell P. J. and R. A. Davis, Time Series: Theory and Appli-cations, Springer-Verlag, New York, (1987).

[5] Cavazos–Cadena R., Computing the asymptotic covariance ma-trix of a vector of sample autocorrelations for ARMA processes ,Applied Mathematics and Computation, 64 (1994), 121–137.

[6] Hoffman K.; R. Kunze, Linear Algebra, Prentice-Hall, EnglewoodCliffs, Massachusetts, (1971).

[7] Lutkepohl H.; Maschke E. O., Bemerkung zur Losung der Yule-Walker-Gleichungen, Metrika, 35 (1988), 287–289.

[8] Martınez–Martınez N. Y.,Implementacion cuadratica del algo-ritmo de innovaciones aplicado a una serie estacionaria, Tesisde Maestrıa en Estadıstica, Direccion de Postgrado, UniversidadAutonoma Agraria Antonio Narro, Saltillo Coah. (2010).

Morfismos -ametaMedotnematrapeDlednoiccudorperedrellatleneemirpmiesticas del Cinvestav, localizado en Avenida Instituto Politecnico Nacional 2508, Colo-

-miedonimretesoremunetsE.F.D,ocixeM,06370.P.C,ocnetacaZordePnaSainprimir en el mes de septiembre de 2012. El tiraje en papel opalina importada de 36kilogramos de 34 × 25.5 cm. consta de 50 ejemplares con pasta tintoreto color verde.

Apoyo tecnico: Omar Hernandez Orozco.

Contents - Contenido

The Mathematical Life of Nikolai Vasilevski

Sergei Grudsky . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

Sistemas de funciones iteradas por partes

Sergio Mabel Juarez Vazquez y Flor de Marıa Correa Romero . . . . . . . . . . . . 9

Uniqueness of the Solution of the Yule–Walker Equations: A Vector SpaceApproach

Ana Paula Isais-Torres and Rolando Cavazos-Cadena . . . . . . . . . . . . . . . . . . . 29