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Topic: Poisson summation formula

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	Poisson summation formula - Wikipedia, the free encyclopedia
		The Poisson summation formula (PSF) is an equation relating a sum S(t) of a function f(t) over all integers and an equivalent summation of its continuous Fourier transform
		The PSF was discovered by Siméon Denis Poisson and is sometimes called Poisson resummation.
		Computationally, the PSF is useful since a slowly converging summation in real space is guaranteed to be converted into a quickly converging equivalent summation in Fourier space.
	en.wikipedia.org /wiki/Poisson_summation_formula (609 words)

	ipedia.com: Simeon Poisson Article (Site not responding. Last check: 2007-10-09)
		Poisson was born at Pithiviers in the département of Loiret, France.
		Poisson was first sent to an uncle, a surgeon at Fontainebleau, and began to take lessons in bleeding and blistering, but made little progress.
		Poisson showed that the result could be extended to a second approximation, and thus made an important advance in the planetary theory.
	www.ipedia.com /simeon_poisson.html (1687 words)

	Ewald summation - Wikipedia, the free encyclopedia
		Ewald summation is a method for computing the interaction energies of crystals, particularly electrostatic energies.
		Ewald summation is a special case of the Poisson summation formula, replacing the summation of interaction energies in real space with an equivalent summation in Fourier space.
		The Ewald summation was developed by Paul Peter Ewald in 1921 (see References below) to determine the electrostatic energy (and, hence, the Madelung constant) of ionic crystals.
	en.wikipedia.org /wiki/Ewald_summation (422 words)

	Jerusalem Mathematics Colloquium (Site not responding. Last check: 2007-10-09)
		In the early 1900's Voronoi conjectured the existence of similar summation formulae for weighted sums of a(n) f(n), where a(n) is an arithmetic quantity.
		His formulae were used to reduce the error term in Gauss' circle problem (the number of lattice points in a circle of area X) from the trivial bound O(X^(1/2)), to O(X^(1/3)).
		Although Voronoi proved his formulae by applying Poisson summation, they are nowadays thought of in terms of modular forms on the complex upper half plane, and L-functions.
	www.ma.huji.ac.il /~colloq/2002-03/col.030109.html (216 words)

	PlanetMath: Poisson summation formula
		The Poisson summation formula is the assertion that
		, we obtain the Poisson summation formula (1).
		This is version 12 of Poisson summation formula, born on 2003-02-11, modified 2005-09-09.
	planetmath.org /encyclopedia/PoissonSummation.html (185 words)

	Selberg trace formula and zeta functions (Site not responding. Last check: 2007-10-09)
		Selberg discovered that the so called "Poisson summation formula" of classical Fourier analysis had a noncommutative generalization that could be applied to obtain an array of important identities in number theory and the theory of automorphic functions.
		This formula, which was motivated by Riemann's zeta function, relates in an exact way the spectrum of the quantal motion on compact surfaces of negative curvature to the classical motion.
		Its divisor is determined by the eigenvalues and scattering poles of the Laplacian and the Euler characteristic of the surface.
	www.maths.ex.ac.uk /~mwatkins/zeta/physics4.htm (4704 words)

	Citebase - Poisson Summation Formula for The Space of Functionals
		Furthermore Using the second infinitesimal and the lattice, we extend the Poisson summation formula of finite group to infinitesimal Fourier transformations for the space of functions and also for the space of functionals.
		Using our Poisson summation formula for functionals, we study a relationship between the functional and the Riemann zeta function.
		Poisson summation formula for infinitesimal Fourier transformation by Kinoshita We extend Poisson summation formula of finite group to Kinoshita's infinitesimal Fourier transformation.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/0405245 (3302 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		In the early 1900s Voronoi conjectured the existence of similar summation formulas for weighted sums of a(n) f(n), where a(n) is an arithmetic quantity.
		His formulas were used to reduce the error term in Gauss' circle problem (the number of lattice points in a circle of area X) from the trivial bound O(X^(1/2)), to O(X^(1/3)).
		Though Voronoi proved his formulae by applying Poisson summation, they are nowadays thought of in terms of modular forms on the complex upper half plane, and L-functions.
	www.math.technion.ac.il /~techm/20030109160020030109mil (248 words)

	Nyquist–Shannon interpolation formula - Wikipedia, the free encyclopedia
		The Nyquist–Shannon interpolation formula or Cardinal series dates back to works of E. Borel in 1898 and was cited from works of J. Whittaker in 1935 in the formulation of the Nyquist–Shannon sampling theorem by C. Shannon in 1949.
		Two derivations of the interpolation formula can be found in the Nyquist-Shannon sampling theorem article, which points out that it can be also be expressed as the convolution of an infinite impulse train with a sinc function:
		This is equivalent to filtering the impulse train with an ideal (brick-wall) low-pass filter.
	en.wikipedia.org /wiki/Nyquist-Shannon_Interpolation_Formula (214 words)

	[No title]
		POISSON SUMMATION FORMULA In this section we first define the Fourier transform in different settings.
		Further, the PSF is a staple in number theory, not only at the level of the Poisson summation formula, but in establishing analytic continuations of zeta functions.
		THE POISSON SUMMATION FORMULA We revisit the PSF since in certain settings it is essentially equivalent to the CST; and the CST can be proved by means of the PSF.
	www.math.umd.edu /~jjb/elemHA.html (1585 words)

	Perspectives in Mathematics Seminar (Site not responding. Last check: 2007-10-09)
		The trace formula, invented in the 1950's by Eichler and Selberg, is one of the most powerful tools for studying automorphic forms.
		It can be viewed as a generalization of the Poisson Summation Formula $$sum_{n=-infty}^{infty} f(n) = sum_{n=-infty}^{infty} hat f(n)$$ Automorphic forms have been studied intensively in recent decades, largely motivated by the deep but still mysterious connections they have withnumber theory.
		The trace formula allows us to do this by computing the trace of certain integral operators acting on spaces of automorphic forms.
	www.math.ucla.edu /~gso/talks/20020429.html (136 words)

	PlanetMath:
		Pascal's formula (in generalized binomial coefficients) owned by pahio
		Poisson distribution (=Poisson random variable) owned by Koro
		Poisson summation (=Poisson summation formula) owned by rmilson
	planetmath.org /encyclopedia/P (2514 words)

	Sampling multipliers and the Poisson Summation Formula (Site not responding. Last check: 2007-10-09)
		Periodization and sampling operators are defined, and the Fourier transform of periodization is uniform sampling in a well-defined sense.
		Implementing this point of view, Poisson Summation Formulas are proved in several spaces including integrable functions of bounded variation (where the result is known) and elements of mixed norm spaces.
		Operators L of this type are appropriately called sampling multipliers; and they give rise to new uniform sampling formulas, where the sampling coefficients of a function f in a reconstruction formula are in terms of the generalized samples Lf.
	www.mat.univie.ac.at /~nuhag/papers/1997/bzi1397.html (176 words)

	A Poisson Summation Formula And Lower Bounds For Resonances In Hyperbolic Manifolds - Perry (ResearchIndex) (Site not responding. Last check: 2007-10-09)
		9 and 1.10 will be used to derive a Poisson formula for scattering resonances for the convex co compact manifolds considered...
		52.6%: A Poisson Summation Formula And Lower Bounds For Resonances In..
		A Poisson summation formula and lower bounds for resonances in hyperbolic manifolds.
	citeseer.ist.psu.edu /377107.html (538 words)

	[No title]
		By Poisson summation formula it is found that f(n) = 1/2+2 NIntegrate[(n-t)^t Cos[Pi t],{t,0,n}] to very good approximation.
		you write: > >By Poisson summation formula it is found that > > I'm with you so far > > >to very good approximation.
		In a follow-up article [Permission pending] wrote: >By Poisson summation formula it is found that > >f(n) = 1/2+2 NIntegrate[(n-t)^t Cos[Pi t],{t,0,n}] > >to very good approximation.
	www.math.niu.edu /~rusin/known-math/95/altern.sum (1264 words)

	D. Bump on the Poisson Summation Formula (Site not responding. Last check: 2007-10-09)
		Although in the explanation above, the Poisson Summation Formula is presented as a straightforward result of Fourier analysis, it is possible to interpret it as a kind of simple "trace formula" on a torus.
		The PSF is seen in this context as relating the spectrum of the Laplacian on a torus to the lengths of its closed geodesics.
		A general genus n compact Riemann surface also has a Laplacian spectrum and a set of shortest path lengths in each deformation class, and it turns out that these sets of values can be related according to an analogous formula, essentially the Selberg trace formula.
	www.maths.ex.ac.uk /~mwatkins/zeta/bump-poisson.htm (153 words)

	Citebase - A Truncated Integral of the Poisson Summation Formula
		Citebase - A Truncated Integral of the Poisson Summation Formula
		A Truncated Integral of the Poisson Summation Formula
		Arthur, A trace formula for reductive groups I: terms associated to classes in G(Q), Duke Math J. Arthur, A measure on the unipotent variety, Canad.
	citebase.eprints.org /cgi-bin/citations?id=oai:arXiv.org:math/9601217 (298 words)

	Poisson's Summation Formula in the Construction of Wavelet Bases (Site not responding. Last check: 2007-10-09)
		Poisson's Summation Formula in the Construction of Wavelet Bases
		In the construction of Multiresolution Analyses for non-Hilbert spaces arising in applications, it is necessary to examine generalized sampling operators, i.e., linear operators of multiplier type from function spaces to sequence spaces.
		It is well known that on the Schwartz space, PSF holds pointwise.
	www.mat.univie.ac.at /~nuhag/papers/1996/bzi1196.html (140 words)

	Poisson summation formula: Encyclopedia topic (Site not responding. Last check: 2007-10-09)
		Its discoverer is Simeon Poisson (Simeon Poisson: more facts about this subject).
		Some conditions restricting F must naturally be applied to have convergence (convergence: The act of converging (coming closer)) here.
		In non-commutative harmonic analysis (harmonic analysis: Analysis of a periodic function into a sum of simple sinusoidal components), the idea is taken even further in the Selberg trace formula (Selberg trace formula: in mathematics, the selberg trace formula is a central result, or area of research, in...
	www.absoluteastronomy.com /reference/poisson_summation_formula (574 words)

	Sharp Embeddings for Modulation Spaces and the Poisson Summation Formula (ResearchIndex) (Site not responding. Last check: 2007-10-09)
		The theorem is closely related to the problem of pointwise convergence in the Poisson summation formula (PSF).
		We also show that this sharp version is optimal in some sense by extending a family of counterexamples to PSF also due to Grochenig.
		3 An uncertainty principle related to the Poisson summation fo..
	citeseer.ist.psu.edu /217699.html (285 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		Subject: Re: summation methods Date: Mon, 07 Jun 1999 13:14:10 GMT Newsgroups: sci.math Keywords: Poisson's summation formula In article
		wrote: > In a certain article I've read that the infinite sum (-1)^n/(n^2+1) can > be evaluate through Poisson's summation formula or Mittag-Leffler > expansions of trigonometric functions.
		Poisson summation: sum_{n=-infinfy}^infinity f(n) = sum_{m=-infinfy}^infinity F(m) where F is the Fourier transform of f: F(t) = integral_{-infinity}^infinity exp(- 2 pi i xt) f(x) dx, provided that f satisfies some mild analytic conditions: for instance that it be differentiable and it and its derivative decay resonably rapidly at infinity.
	www.math.niu.edu /~rusin/known-math/99/poisson_sum (131 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		We use spherical summation of the Fourier series, over (j,k) with j
		A partial explanation for this choppiness rests in the poor convergence one sees in an effort to apply the Poisson summation formula to analyze Fourier inversion on the torus in terms of Fourier inversion on Euclidean space.
		However, for our function R, this choppiness "magically" clears up whenever N is an integral multiple of (2/3) pi.
	www.math.unc.edu /Faculty/met/fourier2.html (333 words)

	[No title] (Site not responding. Last check: 2007-10-09)
		Show that it has a meromorphic continuation to the complex plane.
		(This can be done with the summation formula.) Algebra: Talk about groups of order 55.
		How can this be used to get a formula for the class number?
	www.math.princeton.edu /graduate/generals/vatsal_vinayak (474 words)

	A generalization of Poisson’s summation formula, S. Bochner
		A generalization of Poisson’s summation formula, S. Bochner
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	projecteuclid.org /getRecord?id=euclid.dmj/1077491742 (71 words)

	Amazon.com: An Introduction to the Theory of the Riemann Zeta-Function (Cambridge Studies in Advanced Mathematics): ... (Site not responding. Last check: 2007-10-09)
		Simon (Series Editor) "In this chapter it will be shown how the Poisson Summation Formula leads to the functional equation for the zeta-function..." (more)
		In this chapter it will be shown how the Poisson Summation Formula leads to the functional equation for the zeta-function.
		Riemann Hypothesis, Poisson Summation Formula, Cauchy's Theorem, Prime Number Theorem, Stirling's Formula, Use Exercise, Mean Value Theorem, Euler Product, Jensen's Theorem, Euler's Constant, Fubini's Theorem, Let Show, Lindelbf Hypothesis
	www.amazon.com /exec/obidos/tg/detail/-/0521499054?v=glance (997 words)

	Lecture outline for Wavelet and Multiresolution Analysis
		Fri (2/14): The Poisson and Gauss-Weierstrass kernels and derivation of their Fourier transforms.
		) as convolution operators with, respectively, the Poisson and Gauss-Weierstrass kernels.
		(R) function; The Poisson summation formula for f and its Fourier transform; Jacobi's identity.
	www.math.sc.edu /~sharpley/math758S_sp03/lectures.html (717 words)

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