math research 2014

Mathematical Analysis and Its Applications in theDept. of Math at Tallinn Uni:the past and possible future

Andi Kivinukk

Matemaatika osakond, Tallinna Ülikool

IFI seminar, TLÜNovember 19, 2014

A. Kivinukk (Tallinna Ülikool) 1 / 18

Staff

Staff

Prof Anne Tali (at TLU since 1973)

Mathematical competencies: mathematical analysis, applications offunctional analysis (summability theory)

Prof AK (at TLU since 1993)

Mathematical competencies: mathematical analysis (approximationtheory and its applications, in particular in signal analysis), Fourieranalysis, mathematical finance (option theory), optimization

Senior researcher Maria Zeltser (at TLU since 2004 )

Mathematical competencies: mathematical analysis, applications offunctional analysis, mathematical statistics, data analysis


Staff

Part-time lecturer Anna Šeletski (at TLU since )

Mathematical competencies: mathematical analysis, applications offunctional analysis (summability theory)

Doctoral student Tarmo Metsmägi (defences January, 2015)

Mathematical competencies: mathematical analysis (approximationtheory)

Doctoral student Anna Saksa

Mathematical competencies: mathematical analysis (approximationtheory, Fourier series)


Summability methods, speeds of convergence and ...

Comparison of summability methods, speeds ofconvergence and statistical convergence

Topics by: Anne Tali, her former doctoral student Anna Šeletski,Ulrich Stadtmüller (University of Ulm)

A number sequence x = (ξn) can be convergent or divergent, but onlyconvergent sequences are needed in practice.

A divergent sequence x = (ξn) can be transformed into convergentsequence y = (ηn) by some transformation A. Then it is said thatsequence x is A-convergent, i.e., convergent with respect summabilitymethod (transformation) A.

The most common transformations A are matrix transformationsA = (ank ) defined by

ηn =∞∑

k=0

ankξk , n = 0,1,2, ....


Summability methods, speeds of convergence and ...

The following problems are discussed for certain families {Aα} (α is acontinuous parameter).

1) Methods Aα are compared by their convergence fields (i.e., by thesets of all Aα-convergent sequences) and by speed of convergence.2) The estimates for speeds of methods Aα are found.3) Different types of Aα-convergence, like ordinary convergence,strong convergence and statistical convergence, are charaterized andcompared.4) Transformations Aα are characterized as bounded operators insequence spaces lp.

Anna Šeletski defended her doctoral thesis "Comparison ofsummability methods by summability fields, speeds of convergenceand statistical convergence in a Riesz-type family" in TLU in 2011.


Series and sequences


Topics by

Maria Zeltser

Releasing monotonicity assumption in different tests for convergenceof number series using the WM property. A non-negative tending tozero sequence {ak} weak monotone, written WMS, if for some C > 0it satisfies

ak ≤ Can for any k ∈ [n,2n].

It appears that in Maclaurin-Cauchy integral test, Cauchy condensationtest, Schlömilch and Abel’s k-th term theorem monotonicity can bereplaced by WM. Another generalization of monotonicity allows us togive conditions for a sequence {ak} to satisfy mkak → 0 for a givensequence {mk} tending to infinity.



Another research interest is related to describing sequence spaceswith the help of 0-1 sequences which it contains. The aim is to findconditions in case of different types of sequence spaces E when anygiven sequence space F with a good structure containing all 0-1sequences of the space E contains the space E itself.

Could it be interesting for computer science ??? AK


Approximations, Fourier Analysis, Shannon sampling series

Approximations, Fourier Analysis, Shannon samplingseries

Topics by:

AK and his doctoral students Tarmo Metsmägi and Anna Saksa

Consider a periodic function f ∈ C2π as a signal it can be recovered byits Fourier series or by some generalization

Un(f , x) :=n∑

k=−n

λ(kn)f∧(k)eikx ,

defined by the window functionλ ∈ C[−1,1], λ(0) = 1, λ(u) = 0 (|u| ≥ 1).


Approximations, Fourier Analysis, Shannon sampling series

For non-periodic case the Fourier transform or the Shannon samplingoperators

(SW f )(t) :=∑k∈Z

f (kW

)s(Wt − k)

have used. Here the kernel function is defined by

s(t) :=∫ 1

0λ(u) cos(πtu)du.

Typical problems are how to characterize the error

‖f − SW f‖C .


A selection of Publications


A. Kivinukk, G. Tamberg, On window methods in generalizedShannon sampling operators. In: New Perspectives onApproximation and Sampling Theory. A. I. Zayed and G.Schmeisser (Eds.) Applied and Numerical Harmonic Analysis,Springer, 2014, 65–88.Kivinukk, A. and Metsmägi, T. The variation detracting property ofsome Shannon sampling series and their derivatives. Sampl.Theory Signal Image Process., 13 (2014), no 2, 189–206.A. Kivinukk, On some Shannon sampling series with the variationdetracting property. In Proc. of the 9th Intern. Conf. on SamplingTheory and Applications , Singapore, May 2-6, 2011, A. Khong, F.Oggier (Eds.), Nanyang Techn. Univ., 2011, 1–4



Kivinukk, A. and Metsmägi, T. Approximation in variation by theMeyer-König and Zeller operators. Proc. Estonian Acad. Sci.,2011, 60, 2, 88-97.Kivinukk, A. and Metsmägi, T. Approximation in variation by theKantorovich operators. Proc. Estonian Acad. Sci., 2011, 60, 4,201-209.A. Kivinukk, G. Tamberg, Interpolating generalized Shannonsampling operators, their norms and approximation properties.Sampl. Theory Signal Image Process. 8 (2009) 77–95.



U. Stadtmüller, A. Tali, A family of generalized Nörlund methodsand related power series methods applied to double sequences,Math. Nachr., 2009, 282, 2, 288–306.U. Stadtmüller, A. Tali, A note on families of generalized Nörlundmatrices as bounded operators on lp. Proc. Estonian Acad. Sci.,2009, 58, 3, 137–145.A. Šeletski, A. Tali, Comparison of speeds of convergence inRiesz-type families of summability methods. II, Math. Model.Anal., 2010, 15, 103–112.A. Šeletski, A. Tali, Strong summability methods in a Riesz-typefamily, Proc. Estonian Acad. Sci., 2011, 60, 4, 238–250.A. Šeletski, A. Tali, Comparison of strong and statisticalconvergences in some families of summability methods, Filomat(to appear).



(1.1) M. Zeltser, Bounded domains of generalized Riesz methodswith the Hahn property, Journal of Function Spaces andApplications, 1–8, 2013.(1.1) M. Zeltser, On equiconvergence of number series,Mathematica Slovaca, 63(6), 1333–1346, 2014.(3.1) M. Zeltser, The Hahn property of bounded domains of somematrix methods, AIP Conference Proceedings, 11THINTERNATIONAL CONFERENCE OF NUMERICAL ANALYSISAND APPLIED MATHEMATICS 2013: ICNAAM 2013: Rhodes,Greece, 21-27 Septmeber 2013, 770–773, 2013.(5.2) M. Zeltser, On the Hahn property of bounded domains ofspecial matrix methods, Kangro-100 : Methods of Analysis andAlgebra, Intern. conf. dedicated to the centennial of professorGunnar Kangro, Tartu, Estonia, September 1-6, 2013, Book ofAbstracts, Tartu, Estonian Mathematical Society, 143–143, 2013.



(1.1) M. Zeltser, Factorable Matrices and their associated Rieszmatrices, Proceedings of the Estonian Academy of Sciences.Physics. Mathematics, 63 (4), 1–7, 2014 [to apppear].(3.1) S. Tikhonov, M. Zeltser, Weak Monotonicity Concept and ItsApplications, Fourier Analysis . Pseudo-differential Operators,Time-Frequency Analysis and Partial Differential Equations,Springer, 357– 374, 2014.(5.2) M. Zeltser, Application of weak monotonicity in number seriesand Hardy inequalities. Abstracts of the International Congress ofMathematicians (ICM 2014), Seoul, Korea, 272–273, 2014.(5.2) M. Zeltser, Weak monotonicity concept and its applications,FINEST MATH 2014 : Fourth Finnish-Estonian MathematicsColloquium and Finnish Mathematical Days 2014, Book ofAbstracts, University of Helsinki, 9-10 January 2014, 2014.


Conferences, Lectures, etc.


Fourth Finnish-Estonian Mathematics Colloquium and FinnishMathematical Days 2014, Univ. of Helsinki, 9-10 January 2014.Intern. Conf. on Operator Theory, 28 April - 01 May, 2014,Hammamet, Tunisia.Intern. Congress of Mathematicians (ICM 2014), Seoul, Korea,13-21 August 2014.ISAAC 9th congress, Krakow, Poland, 5-9 august, 2013.Lecturers in Estonian Doctoral School of Mathematics andStatistics , Tartu, 2012, 2013.Conf. Numerical Analysis and Applied Mathematics ICNAAM2013: Rhodes, Greece, 21-27 September 2013.14.03-21.03.2013, Israel, Bar-Ilan University, talk and scientificwork with a co-author.



Modern Time-Frequency Analysis, Strobl, Austria, June 1-7, 2014.10th Intern. Conf. on Sampling Theory and Applications, July 1st -July 5th, 2013, Jacobs Univ. BremenERASMUS lecturer at Babes - Bolyai University of Cluj - Napoca,Romania, March, 2013.3rd Dolomites Workshop on Constructive Approximation andApplications, Alba di Canazei, September 9-14, 2012.


Current Projects

Current projects

Function and sequence spaces in approximations and theirapplications, ETF 8627, 2011 - 2014 (M. Zeltser, AK, TatjanaTamberg, Anna Saksa, Tarmo Metsmägi )

Stochastic processes in nano- and meso-systems : Theory andapplications in material sciences and bio-chemistry, 2012 - 2014(M. Zeltser)


Future: some 2 - 5 years later

Future: some 2 - 5 years later

Retirements, new colleagues with ??? (unknown) qualification

With high probability no pure scientific projects (due to Estonianpolitics in science)

No students, no money, no staff (Now still the situation better thanin physics, biology, ...)


math research 2014

Education