Poisson Summation Research Articles

Voronoi summation formulae on GL(n)

We discover new Voronoi formulae for automorphic forms on GL(n) for n≥4. There are [n/2] different Voronoi formulae on GL(n), which are Poisson summation formulae weighted by Fourier coefficients of the automorphic form with twists by some hyper-Kloosterman sums.

The Weibull distribution and Benford’s law

Benford's law states that many data sets have a bias towards lower leading digits (about $30\%$ are 1s). There are numerous applications, from designing efficient computers to detecting tax, voter and image fraud. It's important to know which common probability distributions are almost Benford. We show the Weibull distribution, for many values of its parameters, is close to Benford's law, quantifying the deviations. As the Weibull distribution arises in many problems, especially survival analysis, our results provide additional arguments for the prevalence of Benford behavior. The proof is by Poisson summation, a powerful technique to attack such problems.

Finite size effect on classical ideal gas revisited

Finite size effects on classical ideal gas are revisited. The micro-canonical partition function for a collection of ideal particles confined in a box is evaluated using Euler–Maclaurin’s as well as Poisson's summation formula. In Poisson's summation formula there are some exponential terms which are absent in Euler–Maclaurin’s formula. In the thermodynamic limit the exponential correction is negligibly small but in the macro/nano dimensions and at low temperatures they may have a great significance. The consequences of finite size effects have been illustrated by redoing the calculations in one and three dimensions keeping the exponential corrections. Global and local thermodynamic properties, diffusion driven by the finite size effect, and effect on speed of sound have been discussed. Thermo-size effects, similar to thermoelectric effects, have been described in detail and may be a theoretical basis with which to design nano-scaled devices. This paper can also be very helpful for undergraduate and graduate students in physics and chemistry as an instructive exercise for a good course in statistical mechanics.

Beyond endoscopy via the trace formula: 1. Poisson summation and isolation of special representations

With analytic applications in mind, in particular beyond endoscopy, we initiate the study of the elliptic part of the trace formula. Incorporating the approximate functional equation into the elliptic part, we control the analytic behavior of the volumes of tori that appear in the elliptic part. Furthermore, by carefully choosing the truncation parameter in the approximate functional equation, we smooth out the singularities of orbital integrals. Finally, by an application of Poisson summation we rewrite the elliptic part so that it is ready to be used in analytic applications, and in particular in beyond endoscopy. As a by product we also isolate the contributions of special representations as pointed out in [Beyond endoscopy, in Contributions to automorphic forms, geometry and number theory (Johns Hopkins University Press, Baltimore, MD, 2004), 611–697].

Fourier Pairs of Discrete Support with Little Structure

We give a simple proof of the fact, first proved in a stronger form in Lev and Olevskii (Quasicrystals with discrete support and spectrum, arXiv preprint arXiv:1501.00085 , 2014), that there exist measures on the real line of discrete support, whose Fourier Transform is also a measure of discrete support, yet this Fourier pair cannot be constructed by repeatedly applying the Poisson Summation Formula finitely many times. More specifically the support of both the measure and its Fourier Transform are not contained in a finite union of arithmetic progressions.

Asymptotics for the Number of Spanning Trees in Circulant Graphs and Degenerating d-Dimensional Discrete Tori

In this paper we obtain precise asymptotics for certain families of graphs, namely circulant graphs and degenerating discrete tori. The asymptotics contain interesting constants from number theory among which some can be interpreted as corresponding values for continuous limiting objects. We answer one question formulated in a paper from Atajan, Yong, and Inaba in [1] and formulate a conjecture in relation to the paper from Zhang, Yong, and Golin [23]. A crucial ingredient in the proof is to use the matrix tree theorem and express the combinatorial Laplacian determinant in terms of Bessel functions. A non-standard Poisson summation formula and limiting properties of theta functions are then used to evaluate the asymptotics.

Operator-Valued Frames for the Heisenberg Group

A classical result of Duffin and Schaeffer gives conditions under which a discrete collection of characters on \({\mathbb {R}}\), restricted to \(E=(-\gamma ,\gamma )\subsetneq (-1/2,1/2)\), forms a Hilbert-space frame for \(L^2(E)\). For the case of characters with period one, this is just the Poisson Summation Formula. Duffin and Schaeffer show that perturbations preserve the frame condition in this case. This paper gives analogous results for the real Heisenberg group \(H_n\), where frames are replaced by operator-valued frames. The Selberg Trace Formula is used to show that perturbations of the orthogonal case continue to behave as operator-valued frames. This technique enables the construction of decompositions of elements of \(L^2(E)\) for suitable subsets \(E\) of \(H_n\) in terms of representations of \(H_n\).

On the Duality of Discrete and Periodic Functions

Although versions of Poisson’s Summation Formula (PSF) have already been studied extensively, there seems to be no theorem that relates discretization to periodization and periodization to discretization in a simple manner. In this study, we show that two complementary formulas, both closely related to the classical Poisson Summation Formula, are needed to form a reciprocal Discretization-Periodization Theorem on generalized functions. We define discretization and periodization on generalized functions and show that the Fourier transform of periodic functions are discrete functions and, vice versa, the Fourier transform of discrete functions are periodic functions.

Computing the truncated theta function via Mordell integral

Hiary [3] has presented an algorithm which allows to evaluate the truncated theta function P n k=0 exp(2 i(zk +k 2 )) to within in O(ln( n ) ) arithmetic operations for any real z and . This remarkable result has many applications in Number Theory, in particular it is the crucial element in Hiary’s algorithm for computing ( 1 + it) to within t in O (t 1 3 ln(t) ) arithmetic operations, see [2]. We present a signicant simplication of Hiary’s algorithm for evaluating the truncated theta function. Our method avoids the use of the Poisson summation formula, and substitutes it with an explicit identity involving the Mordell integral. This results in an algorithm which is ecient, conceptually simple and easy to implement.

Fast Calculation of Scattering by 3-D Inhomogeneities in Uniaxial Anisotropic Multilayers

A fast algorithm to calculate the response of uniaxial layered media on a rectilinear mesh is proposed in order to handle scattering phenomena due to inhomogeneities within them. The uniaxial layered media are with their optical axes parallel to the interfaces of the layered structure. The generalized Poisson summation formula is used to efficiently calculate the discrete Fourier transform of spatial responses of the layeredmedia from its continuous Fourier spectrum. Using the windowing technique, the integral involved in constructing the discrete Fourier spectrum can be performed in a small domain without compromising the accuracy of the result. Furthermore, a fast numerical integration method based on the recently proposed Padua points is used, which, together with the windowing technique, yields a fast solution to such scattering problems. Numerical simulations illustrate the accuracy and efficiency of the proposed algorithm.

Poisson summation for functions of bounded variation on ℝd

Poisson summation for functions of bounded variation on ℝd

Seven pivotal theorems of Fourier analysis, signal analysis, numerical analysis and number theory: their interconnections

The present paper deals mainly with seven fundamental theorems of mathematical analysis, numerical analysis, and number theory, namely the generalized Parseval decomposition formula (GPDF), introduced 15 years ago, the well-known approximate sampling theorem (ASF), the new approximate reproducing kernel theorem, the basic Poisson summation formula, already known to Gauß, a newer version of the GPDF having a structure similar to that of the Poisson summation formula, namely, the Parseval decomposition–Poisson summation formula, the functional equation of Riemann’s zeta function, as well as the Euler–Maclaurin summation formula. It will in fact be shown that these seven theorems are all equivalent to one another, in the sense that each is a corollary of the others. Since these theorems can all be deduced from each other, one of them has to be proven independently in order to verify all. It is convenient to choose the ASF, introduced in 1963. The epilogue treats possible extensions to the more general contexts of reproducing kernel theory and of abstract harmonic analysis, using locally compact abelian groups. This paper is expository in the sense that it treats a number of mathematical theorems, their interconnections, their equivalence to one another. On the other hand, the proofs of the many intricate interconnections among these theorems are new in their essential steps and conclusions.

Estimates of three classical summations on the spaces $\dot F_p^{\alpha ,q} (\mathbb{R}^n )(0 < p \leqslant 1)$

We obtain the boundedness on \(\dot F_p^{\alpha ,q} (\mathbb{R}^n )(0 < p \leqslant 1)\) for the Poisson summation and Gauss summation. Their maximal operators are proved to be bounded from \(\dot F_p^{\alpha ,q} (\mathbb{R}^n )(0 < p \leqslant 1)\) to \(L^\infty (\mathbb{R}^n )\). For the maximal operator of the Bochner-Riesz summation, we prove that it is bounded from \(\dot F_p^{\alpha ,q} (\mathbb{R}^n )(0 < p \leqslant 1)\) to \(L^{\tfrac{{pn}} {{n - p\alpha }},\infty } (\mathbb{R}^n ) \).

Coupling of gravity to matter, spectral action and cosmic topology

We consider a model of modified gravity based on the spectral action functional, for a cosmic topology given by a spherical space form, and the associated slow-roll inflation scenario. We consider then the coupling of gravity to matter determined by an almost-commutative geometry over the spherical space form. We show that this produces a multiplicative shift of the amplitude of the power spectra for the density fluctuations and the gravitational waves, by a multiplicative factor equal to the total number of fermions in the matter sector of the model. We obtain the result by an explicit nonperturbative computation, based on the Poisson summation formula and the spectra of twisted Dirac operators on spherical space forms, as well as, for more general spacetime manifolds, using a heat kernel computation.

Periodized radial basis functions, part I: Theory

We extend the theory of periodized RBFs. We show that the imbricate series that define the Periodic Gaussian (PGA) and Sech (PSech) basis functions are Jacobian theta functions and elliptic functions “dn”, respectively. The naive periodization fails for the Multiquadric and Inverse Multiquadric RBFs, but we are able to define periodic generalizations of these, too, by proving and exploiting a generalization of the Poisson Summation Theorem. Although applications of periodic RBFs are mostly left for another day, we do illustrate the flaws and potential by solving the Mathieu eigenproblem on both uniform and highly-adapted grids. The terms of a Fourier basis can be grouped into four classes, depending upon parity with respect to both the origin and x=π/2, and so, too, the Mathieu eigenfunctions. We show how to construct symmetrized periodic RBFs and illustrate these by solving the Mathieu problem using only the periodic RBFs of the same symmetry class as the targeted eigenfunctions. We also discuss the relationship between periodic RBFs and trigonometric polynomials with the aid of an explicit formula for the nonpolynomial part of the Periodic Inverse Quadratic (PIQ) basis functions. We prove that the rate of convergence for periodic RBFs is geometric, that is, the error can be bounded by exp⁡(−Nμ) for some positive constant μ. Lastly, we prove a new theorem that gives the periodic RBF interpolation error in Fourier coefficient space. This is applied to the “spectral-plus” question. We find that periodic RBFs are indeed sometimes orders of magnitude more accurate than trigonometric interpolation even though it has long been known that RBFs (periodic or not) reduce to the corresponding classical spectral method as the RBF shape parameter goes to 0. However, periodic RBFs are “spectral-plus” only when the shape parameter α is adaptively tuned to the particular f(x) being approximated and even then, only when f(x) satisfies a symmetry condition.

Truncation Diffraction Phenomena of Floquet Waves Radiated From Semi-Infinite Phased Array Antenna in a General Focus Problem

This paper presents uniform formulations of radiation mechanisms, analogous to the ray decompositions of diffraction theory, from a semi-infinite and periodic array of antennas. The antenna array is excited to radiate the electromagnetic (EM) fields focused at a relatively arbitrarily selected location. The ray decomposition results from a closed-form formulation by asymptotically evaluating the radiation integral of Floquet modes that are obtained by applying the Poisson sum formula to the summation of elemental radiations. The analysis is relatively general and will reduce to the case of a conventional phased array antenna that is excited to radiate directive beams focused in the far zone, and exhibits consistent phenomena. Theoretical investigations as well as numerical examinations are presented to demonstrate the radiation mechanisms.

Unequally Spaced Arrays Based on the Poisson Sum Formula

It has been recognized that unequally spaced arrays can offer some advantages over equally spaced arrays, mainly in the following areas: 1) thinned array by reducing the number of elements to achieve the high resolution without producing grating lobes; 2) broadbanding or wide scanning without grating lobes; 3) reducing sidelobes for the array of uniform amplitude; and 4) unequally spaced arrays on a curved surface. Historical accounts and a basic theory of unequally spaced arrays based on Poisson Sum Formula are given. Most of the analytical approach can be obtained by numerical computation. Analytical methods may show additional physical mechanisms and offer alternative approaches. This paper also discusses some additional applications on circular arrays, active impedance and arrays on a curved surface.

On the arithmetic Walsh coefficients of Boolean functions

We generalize to the arithmetic Walsh transform (AWT) some results which were previously known for the Walsh---Hadamard transform of Boolean functions. We first generalize the classical Poisson summation formula to the AWT. We then define a generalized notion of resilience with respect to an arbitrary statistical measure of Boolean functions. We apply the Poisson summation formula to obtain a condition equivalent to resilience for one such statistical measure. Last, we show that the AWT of a large class of Boolean functions can be expressed in terms of the AWT of a Boolean function of algebraic degree at most three in a larger number of variables.

Asymptotic Floquet Mode Investigation Over Two-Dimensional Infinite Array Antennas Phased to Radiate Near-Zone Focused Fields

This paper extends the previous work in to consider the radiation from an infinite, 2-D antenna array for the near-field focus applications. It employs the Poisson sum formula to expand the radiation into Floquet modes, which may exhibit a class of wave phenomena in terms of geometrical ray fields. This analysis automatically reduces to the conventional case of a linearly phased antenna array, and results in consistent radiation phenomena. It can be considered as a generalization of phased array analysis. Theoretical investigations as well as numerical illustrations are presented.

Multifractal behavior of polynomial Fourier series

We study the spectrum of singularities of a family of Fourier series with polynomial frequencies, in particular we prove that they are multifractal functions. The case of degree two was treated by S. Jaffard in 1996. Higher degrees require completely different ideas essentially because harmonic analysis techniques (Poisson summation) are useless to study the oscillation at most of the points. We introduce a new approach involving special diophantine approximations with prime power denominators and fine analytic and arithmetic aspects of the estimation of exponential sums to control the Hölder exponent in thin Cantor-like sets.