Summability methods pFrom Wikipedia, the free encyclopediaContents1 Perrons formula 11.1 Statement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.4 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 Poisson summation formula 32.1 Forms of the equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Distributional formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.3 Derivation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.4 Applicability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.5 Applications. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.5.1 Method of images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.5.2 Sampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.5.3 Ewald summation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.5.4 Lattice points in a sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52.5.5 Number theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.6 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.6.1 Selberg trace formula. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.7 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.9 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.10Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.11Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 82.11.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.11.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.11.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8iChapter 1Perrons formulaIn mathematics, and more particularly in analytic number theory, Perrons formula is a formula due to Oskar Perronto calculate the sum of an arithmetical function, by means of an inverse Mellin transform.1.1 StatementLet {a(n)} be an arithmetic function, and letg(s) =n=1a(n)nsbe the corresponding Dirichlet series. Presume the Dirichlet series to be uniformly convergent for (s) > . ThenPerrons formula isA(x) =nxa(n) =12ic+icig(z)xzzdz.Here, the prime on the summation indicates that the last term of the sum must be multiplied by 1/2 when x is aninteger. The integral is not a convergent Lebesgue integral, it is understood as the Cauchy principal value. Theformula requires c > 0, c > , and x > 0 real, but otherwise arbitrary.1.2 ProofAn easy sketch of the proof comes from taking Abels sum formulag(s) =n=1a(n)ns= s0A(x)x(s+1)dx.This is nothing but a Laplace transform under the variable change x = et. Inverting it one gets Perrons formula.1.3 ExamplesBecause of its general relationship to Dirichlet series, the formula is commonly applied to many number-theoreticsums. Thus, for example, one has the famous integral representation for the Riemann zeta function:12 CHAPTER 1. PERRONS FORMULA(s) = s1xxs+1dxand a similar formula for Dirichlet L-functions:L(s, ) = s1A(x)xs+1dxwhereA(x) =nx(n)and (n) is a Dirichlet character. Other examples appear in the articles on the Mertens function and the von Mangoldtfunction.1.4 ReferencesPage 243 of Apostol, Tom M. (1976), Introduction to analytic number theory, Undergraduate Texts in Mathe-matics, New York-Heidelberg: Springer-Verlag, ISBN 978-0-387-90163-3, MR 0434929, Zbl 0335.10001Weisstein, Eric W., Perrons formula, MathWorld.Tenebaum, Grald (1995). Introduction to analytic and probabilistic number theory. Cambridge Studies inAdvanced Mathematics46. Translated by C.B. Thomas. Cambridge: Cambridge University Press. ISBN0-521-41261-7. Zbl 0831.11001.Chapter 2Poisson summation formulaIn mathematics, the Poisson summation formula is an equation that relates the Fourier series coecients of theperiodic summation of a function to values of the functions continuous Fourier transform. Consequently, the periodicsummation of a function is completely dened by discrete samples of the original functions Fourier transform. Andconversely, the periodic summation of a functions Fourier transform is completely dened by discrete samples of theoriginal function. The Poisson summation formula was discovered by Simon Denis Poisson and is sometimes calledPoisson resummation.2.1 Forms of the equationFor appropriate functions f, the Poisson summation formula may be stated as:With the substitution, g(xP) def=f(x),and the Fourier transform property, F{g(xP)} =1P g_P_(for P > 0),Eq.1 becomes:With another denition, s(t +x) def= g(x), and the transform property F{s(t +x)} = s() ei2t, Eq.2 becomesa periodic summation (with period P) and its equivalent Fourier series:Similarly, the periodic summation of a functions Fourier transform has this Fourier series equivalent:where T represents the time interval at which a function s(t) is sampled, and 1/T is the rate of samples/sec.2.2 Distributional formulationThese equations can be interpreted in the language of distributions (Crdoba 1988; Hrmander 1983, 7.2) for afunction f whose derivatives are all rapidly decreasing (see Schwartz function).Using the Dirac comb distributionand its Fourier series:34 CHAPTER 2. POISSON SUMMATION FORMULAIn other words, the periodization of a Dirac delta , resulting in a Dirac comb, corresponds to the discretization ofits spectrum which is constantly one. Hence, this again is a Dirac comb but with reciprocal increments.Eq.1 readily follows:k=f(k) =k=_f(x) ei2kxdx_=f(x)_k=ei2kx_. .n=(xn)dx=n=_f(x) (x n) dx_=n=f(n).Similarly:k= s( + k/T) =k=F_s(t) ei2kT t_= F_s(t)k=ei2kT t. .Tn=(tnT)_= F_n=T s(nT) (t nT)_=n=T s(nT) F {(t nT)} =n=T s(nT) ei2nT.2.3 DerivationWe can also prove that Eq.3 holds in the sense that if s(t) L1(R), then the right-hand side is the (possibly divergent)Fourier series of the left-hand side. This proof may be found in either (Pinsky 2002) or (Zygmund 1968). It followsfrom the dominated convergence theorem that sP(t) exists and is nite for almost every t. And furthermore it followsthat sP is integrable on the interval [0,P]. The right-hand side of Eq.3 has the form of a Fourier series. So it issucient to show that the Fourier series coecients of sP(t) are1P s(kP ). . Proceeding from the denition of theFourier coecients we have:S[k] def=1PP0sP(t) ei2kP tdt=1PP0_n=s(t + nP)_ ei2kP tdt=1Pn=P0s(t + nP) ei2kP tdt,where the interchange of summation with integration is once again justied by dominated convergence.With a change of variables ( = t + nP) this becomes:S[k] =1Pn=nP+PnPs() ei2kP ei2kn. .1d =1Ps() ei2kP d=1P s_kP_QED.2.4 ApplicabilityEq.3 holds provided s(t) is a continuous integrable function which satises2.5. APPLICATIONS 5|s(t)| + | s(t)| C(1 + |t|)1for some C, > 0 and every t (Grafakos 2004; Stein & Weiss 1971). Note that such s(t) is uniformly continuous,this together with the decay assumption on s, show that the series dening sP converges uniformly to a continuousfunction. Eq.3 holds in the strong sense that both sides converge uniformly and absolutely to the same limit (Stein &Weiss 1971).Eq.3 holds in a pointwise sense under the strictly weaker assumption that s has bounded variation and2 s(t) = lim0 s(t + ) + lim0 s(t ) (Zygmund 1968).The Fourier series on the right-hand side of Eq.3 is then understood as a (conditionally convergent) limit of symmetricpartial sums.As shown above, Eq.3 holds under the much less restrictive assumption that s(t) is in L1(R), but then it is necessaryto interpret it in the sense that the right-hand side is the (possibly divergent) Fourier series of sP(t) (Zygmund 1968).In this case, one may extend the region where equality holds by considering summability methods such as Cesrosummability. When interpreting convergence in this way Eq.2 holds under the less restrictive conditions that g(x)is integrable and 0 is a point of continuity of gP(x). However Eq.2 may fail to hold even when both gand g areintegrable and continuous, and the sums converge absolutely (Katznelson 1976).2.5 Applications2.5.1 Method of imagesIn partial dierential equations, the Poisson summation formula provides a rigorous justication for the fundamentalsolution of the heat equation with absorbing rectangular boundary by the method of images. Here the heat kernelon R2is known, and that of a rectangle is determined by taking the periodization. The Poisson summation formulasimilarly provides a connection between Fourier analysis on Euclidean spaces and on the tori of the correspondingdimensions (Grafakos 2004). In one dimension, the resulting solution is called a theta function.2.5.2 SamplingIn the statistical study of time-series, if f is a function of time, then looking only at its values at equally spaced pointsof time is called sampling. In applications, typically the function f is band-limited, meaning that there is some cutofrequency fo such that the Fourier transform is zero for frequencies exceeding the cuto:f()=0 for || >fo. For band-limited functions, choosing the sampling rate 2fo guarantees that no information is lost: sincef can bereconstructed from these sampled values, then, by Fourier inversion, so can f . This leads to the NyquistShannonsampling theorem (Pinsky 2002).2.5.3 Ewald summationComputationally, the Poisson summation formula is useful since a slowly converging summation in real space isguaranteed to be converted into a quickly converging equivalent summation in Fourier space. (A broad functionin real space becomes a narrow function in Fourier space and vice versa.) This is the essential idea behind Ewaldsummation.2.5.4 Lattice points in a sphereThe Poisson summation formula may be used to derive Landaus asymptotic formula for the number of lattice pointsin a large Euclidean sphere. It can also be used to show that if an integrable function, fandf both have compactsupport then f= 0 (Pinsky 2002).6 CHAPTER 2. POISSON SUMMATION FORMULA2.5.5 Number theoryIn number theory, Poisson summation can also be used to derive a variety of functional equations including thefunctional equation for the Riemann zeta function.[1]One important such use of Poisson summation concerns theta functions:periodic summations of Gaussians . Putq= ei, for a complex number in the upper half plane, and dene the theta function:() =n qn2.The relation between (1/) and () turns out to be important for number theory, since this kind of relation isone of the dening properties of a modular form. By choosingf =ex2in the second version of the Poissonsummation formula (with a = 0 ), and using the fact thatf= e2, one gets immediately_1_=i ()by putting 1/ =/i .It follows from this that 8has a simple transformation property under 1/ and this can be used to proveJacobis formula for the number of dierent ways to express an integer as the sum of eight perfect squares.2.6 GeneralizationsThe Poisson summation formula holds in Euclidean space of arbitrary dimension. Let be the lattice in Rdconsistingof points with integer coordinates; is the character group, or Pontryagin dual, of Rd. For a function in L1(Rd),consider the series given by summing the translates of by elements of :f(x + ).Theorem For in L1(Rd), the above series converges pointwise almost everywhere, and thus denes a periodicfunction P on . P lies in L1() with ||P||1 ||||1. Moreover, for all in , P() (Fourier transform on )equals () (Fourier transform on Rd).When is in addition continuous, and both and decay suciently fast at innity, then one can invert the domainback to Rdand make a stronger statement. More precisely, if|f(x)| + | f(x)| C(1 + |x|)dfor some C, > 0, then f(x + ) =f()e2ix, (Stein & Weiss 1971, VII 2)where both series converge absolutely and uniformly on . When d = 1 and x = 0, this gives the formula given in therst section above.More generally, a version of the statement holds if is replaced by a more general lattice in Rd. The dual lattice can be dened as a subset of the dual vector space or alternatively by Pontryagin duality. Then the statement is thatthe sum of delta-functions at each point of , and at each point of , are again Fourier transforms as distributions,subject to correct normalization.This is applied in the theory of theta functions, and is a possible method in geometry of numbers. In fact in morerecent work on counting lattice points in regions it is routinely used summing the indicator function of a region Dover lattice points is exactly the question, so that the LHS of the summation formula is what is sought and the RHSsomething that can be attacked by mathematical analysis.2.6.1 Selberg trace formulaFurther generalisation to locally compact abelian groups is required in number theory. In non-commutative harmonicanalysis, the idea is taken even further in the Selberg trace formula, but takes on a much deeper character.2.7. SEE ALSO 7A series of mathematicians applying harmonic analysis to number theory, most notably Martin Eichler, Atle Selberg,Robert Langlands, and James Arthur, have generalised the Poisson summation formula to the Fourier transform onnon-commutative locally compact reductive algebraic groups G with a discrete subgroup such that G/ has nitevolume. For example, G can be the real points of GLn and can be the integral points of GLn . In this setting,G plays the role of the real number line in the classical version of Poisson summation, and plays the role of theintegers n that appear in the sum. The generalised version of Poisson summation is called the Selberg Trace Formula,and has played a role in proving many cases of Artins conjecture and in Wiless proof of Fermats Last Theorem.The left-hand side of (1) becomes a sum over irreducible unitary representations of G , and is called the spectralside, while the right-hand side becomes a sum over conjugacy classes of , and is called the geometric side.The Poisson summation formula is the archetype for vast developments in harmonic analysis and number theory.2.7 See alsoFourier_analysis#SummaryPosts inversion formula2.8 Notes[1]f() def=f(x) e2ixdx.2.9 References[1] H. M. Edwards (1974). Riemanns Zeta Function. Academic Press. ISBN 0-486-41740-9. (pages 209-211)2.10 Further readingBenedetto, J.J.;Zimmermann, G. (1997), Sampling multipliers and the Poisson summation formula, J.Fourier Ana. App. 3 (5).Crdoba, A., La formule sommatoire de Poisson, C.R. Acad. Sci. Paris, Series I 306: 373376.Gasquet, Claude; Witomski, Patrick (1999), Fourier Analysis and Applications, Springer, pp. 344352, ISBN0-387-98485-2.Grafakos, Loukas (2004), Classical and Modern Fourier Analysis, Pearson Education, Inc., pp. 253257, ISBN0-13-035399-X.Higgins, J.R. (1985), Five short stories about the cardinal series, Bull. AMS 12 (1): 4589, doi:10.1090/S0273-0979-1985-15293-0.Hrmander, L. (1983), The analysis of linear partial dierential operators I, Grundl. Math. Wissenschaft.256, Springer, ISBN 3-540-12104-8, MR 0717035.Katznelson, Yitzhak (1976), An introduction to harmonic analysis (Second corrected ed.), New York: DoverPublications, Inc, ISBN 0-486-63331-4Pinsky, M. (2002), Introduction to Fourier Analysis and Wavelets., Brooks Cole.Stein, Elias; Weiss, Guido (1971), IntroductiontoFourierAnalysisonEuclideanSpaces, Princeton, N.J.:Princeton University Press, ISBN 978-0-691-08078-9.Zygmund, Antoni (1968), Trigonometric series (2nd ed.), Cambridge University Press (published 1988), ISBN978-0-521-35885-9.8 CHAPTER 2. POISSON SUMMATION FORMULA2.11 Text and image sources, contributors, and licenses2.11.1 Text PerronsformulaSource: https://en.wikipedia.org/wiki/Perron{}s_formula?oldid=637793705Contributors: XJaM, TakuyaMurata,Skysmith, Charles Matthews, Jitse Niesen, Giftlite, Bender235, EmilJ, Oleg Alexandrov, Linas, JosephSilverman, Twold, A. 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