Poisson Summation Research Articles

Photon polarization tensor in a magnetized plasma system

We investigate the photon polarization tensor at finite temperatures in the presence of a static and homogeneous external magnetic field. In our scheme, the summing of the Matsubara frequency is performed after Poisson resummation, which is easily completed and converges quickly. Moreover, the behaviors of finite Landau levels are presented explicitly. It shows a convergence while summing infinite Landau levels. Consequently, there is no necessity to truncate the Landau level in a numerical estimation. At zero temperature, the lowest Landau level (LLL) approximation is analytically satisfied for the vacuum photon polarization tensor. However, we examine that the LLL approximation is not enough for the thermal polarization tensor. The thermal tensor obtains non-trivial contributions from the finite-n Landau levels. And, photon spectra gains a large imaginary contribution in thermal medium, which is the so-called Landau damping. Finally, it is argued that the summation of Matsubara frequency is not commuted with Landau level ones, such conjecture is excluded in our calculations.

Vibro-acoustic characteristics of eccentrically stiffened functionally graded material sandwich cylindrical shell under external mean fluid

The main objective of this research work is focused on acoustic radiation analysis of functionally graded material (FGM) sandwich cylindrical shell structure reinforced by periodically eccentrically ring stiffeners. In the analysis, two common types of FGM sandwich cylindrical shell are considered, which include one with FGM face shell and homogeneous ceramic or metal core, and the other with FGM core and homogeneous face shell. Given the structure is immersed in a mean flowing fluid of velocity tangential to the accoustically deformed boundary and it is excited by a radial point force. The ring stiffeners are assumed to be identical and uniformly spaced, and the in-plane flexural and out-of-plane torsional motions of the ring stiffeners are employed to accurately describe the force moment coupling between the stiffeners and the shell. Fluid-structure coupling is considered by imposing velocity continuity condition at fluid-structure interfaces. By applying the Poisson summation formula and the Fourier transformation technique for periodic ring stiffeners FGM shell structures, the sound pressure level (SPL) about structural acoustic radiation performance index is expressed in a superposition form of space harmonics for a given wave number. The validity of the proposed theoretical model is verified by comparisons with previously published numerical results. Based on the developed theoretical mode, the influences of the FGM properties, gradient index, sandwich type, the size of ring stiffener, external mean flow and skin-core-skin ratios type on the structural acoustic radiation performance have been investigated. In addition, the results show that under the same skin-core-skin ratio and thickness, the acoustic radiation performances of type C seem better, and the larger the section size of ring stiffened, the greater the influence of structure acoustic radiation characteristics, whereas the magnitudes of the SPL curves increase with the increasing values of Mach number.

The density of rational points near hypersurfaces

We establish a sharp asymptotic formula for the number of rational points up to a given height and within a given distance from a hypersurface. Our main innovation is a bootstrap method that relies on the synthesis of Poisson summation, projective duality, and the method of stationary phase. This has surprising applications to counting rational points lying on the manifold; indeed, we are able to prove an analogue of Serre’s dimension growth conjecture (originally stated for projective varieties) in this general setup. As another consequence of our main counting result, we solve the generalized Baker–Schmidt problem in the simultaneous setting for hypersurfaces.

Discrete aspects of continuous symmetries in the tensorial formulation of Abelian gauge theories

We show that standard identities and theorems for lattice models with $U(1)$ symmetry get re-expressed discretely in the tensorial formulation of these models. We explain the geometrical analogy between the continuous lattice equations of motion and the discrete selection rules of the tensors. We construct a gauge-invariant transfer matrix in arbitrary dimensions. We show the equivalence with its gauge-fixed version in a maximal temporal gauge and explain how a discrete Gauss's law is always enforced. We propose a noise-robust way to implement Gauss's law in arbitrary dimensions. We reformulate Noether's theorem for global, local, continuous or discrete Abelian symmetries: for each given symmetry, there is one corresponding tensor redundancy. We discuss semi-classical approximations for classical solutions with periodic boundary conditions in two solvable cases. We show the correspondence of their weak coupling limit with the tensor formulation after Poisson summation. We briefly discuss connections with other approaches and implications for quantum computing.

Functorial transfer between relative trace formulas in rank 1

According to the Langlands functoriality conjecture, broadened to the setting of spherical varieties (of which reductive groups are special cases), a map between L -groups of spherical varieties should give rise to a functorial transfer of their local and automorphic spectra. The “beyond endoscopy” proposal predicts that this transfer will be realized as a comparison between limiting forms of the (relative) trace formulas of these spaces. In this paper, we establish the local transfer for the identity map between L -groups, for spherical affine homogeneous spaces X = H \ G whose dual group is SL 2 or PGL 2 (with G and H split). More precisely, we construct a transfer operator between orbital integrals for the ( X × X ) / G -relative trace formula, and orbital integrals for the Kuznetsov formula of PGL 2 or SL 2 . Besides the L -group, another invariant attached to X is a certain L -value, and the space of test measures for the Kuznetsov formula is enlarged to accommodate the given L -value. The transfer operator is given explicitly in terms of Fourier convolutions, making it suitable for a global comparison of trace formulas by the Poisson summation formula, hence for a uniform proof, in rank 1 , of the relations between periods of automorphic forms and special values of L -functions.

On Binomial and Poisson Sums Arising from the Displacement of Randomly Placed Sensors

We re-visit the asymptotics of a binomial and a Poisson sum which arose as (average) displacement costs when moving randomly placed sensors to anchor positions. The first-order asymptotics of these sums were derived in several stages in a series of recent papers. In this paper, we give a unified approach based on the classical Laplace method with which one can also derive more terms in the asymptotic expansions. Moreover, in a special case, full asymptotic expansions can be given which even hold as identities. This will be proved by a combinatorial approach and systematic ways of computing all coefficients of these identities will be discussed as well.

On Sampling and Interpolation by Model Sets

We refine a result of Matei and Meyer on stable sampling and stable interpolation for simple model sets. Our setting is model sets in locally compact abelian groups and Fourier analysis of unbounded complex Radon measures as developed by Argabright and de Lamadrid. This leads to a refined version of the underlying model set duality between sampling and interpolation. For rather general model sets, our methods also yield an elementary proof of stable sampling and stable interpolation sufficiently far away from the critical density, which is based on the Poisson Summation Formula.

Pure point measures with sparse support and sparse Fourier–Bohr support

Fourier-transformable Radon measures are called doubly sparse when both the measure and its transform are pure point measures with sparse support. Their structure is reasonably well understood in Euclidean space, based on the use of tempered distributions. Here, we extend the theory to second countable, locally compact Abelian groups, where we can employ general cut and project schemes and the structure of weighted model combs, along with the theory of almost periodic measures. In particular, for measures with Meyer set support, we characterise sparseness of the Fourier--Bohr spectrum via conditions of crystallographic type, and derive representations of the measures in terms of trigonometric polynomials. More generally, we analyse positive definite, doubly sparse measures in a natural cut and project setting, which results in a Poisson summation type formula.

The vibroacoustic analysis of periodic structure-stiffened plates

A semi-analytical method is proposed to estimate the vibroacoustic properties of a fluid-loaded thin plate, which is assumed to have an infinite extent and is stiffened by periodically spaced structures in orthogonal directions. By applying the finite element method, the interaction force between the plate and each periodic structure is approximated by a finite number of equivalent nodal forces, which are subsequently represented as functions of the corresponding nodal displacements of the plate. With the space harmonic method and Poisson's summation formula, the acoustic equation of the periodically stiffened plate, stimulated by convected harmonic pressure, is then derived and expressed as a periodic set of nodal displacements. In particular, an efficient method is presented for the vibration response of rib-stiffened plates, which significantly simplifies the solution process. The vibroacoustic properties of different types of periodically stiffened plates under convected harmonic pressures are examined by means of numerical examples. The results demonstrate that, owing to the attachment of the periodic structures to the plate, the sound radiation from the plate system under convected harmonic pressures can be suppressed at low frequencies. The present method can be applied for estimating the vibroacoustic properties of various periodically stiffened plates, which may also contribute to structural design for vibration or sound radiation reduction.

Thermodynamic properties of improved deformed exponential-type potential (IDEP) for some diatomic molecules

Within the framework of non-relativistic quantum mechanics, the ro-vibrational energy spectra of the improved deformed exponential-type potential model are obtained using the Greene-Aldrich approximation scheme and an appropriate coordinate transformation. With the help of the energy spectra, analytical expressions of the vibrational partition function and other thermodynamic functions are derived by employing the Poisson summation formula. These thermodynamic functions are studied for the electronic states of hydrogen dimer, carbon monoxide, nitrogen dimer and lithium hydride diatomic molecules, as they vary with temperature and upper bound vibration quantum number.

Thermal properties of Deng–Fan–Eckart potential model using Poisson summation approach

The Deng-Fan-Eckart (DFE) potential is as good as the Morse potential in studying atomic interaction in diatomic molecules. By using the improved Pekeris-type approximation, to deal with the centrifugal term, we obtain the bound-state solutions of the radial Schr\"odinger equation with this adopted molecular model via the Factorization Method. With the energy equation obtained, the thermodynamic properties of some selected diatomic molecules(H2 , CO , and ScN ) were obtained using Poisson summation method.. The unnormalized wave function is also derived. The energy spectrum for a set of diatomic molecules for different values of the vibrational n and rotational l are obtained. To show the accuracy of our results, we discuss some special cases by adjusting some potential parameters and also compute the numerical eigenvalue of the Deng-Fan potential for comparison sake. However, it was found out that our results agree excellently with the results obtained via other methods.

Optimal Truncations for Multivariate Fourier and Chebyshev Series: Mysteries of the Hyperbolic Cross: Part I: Bivariate Case

The key to most successful applications of Chebyshev and Fourier spectral methods in high space dimension are a combination of a Smolyak sparse grid together with so-called “hyperbolic cross” truncation. It is easy to find counterexamples for which the hyperbolic cross truncation is far from optimal. An important question is: what characteristics of a function make it “crossy”, that is, suitable for the hyperbolic crosss truncation? We have not been able to find a complete answer to this question. However, by combining low-rank SVD approximation, Poisson summation and imbricate series, hyperbolic coordinates and numerical experimentation, we are, to borrow from Fermi, “confused at a higher level”. For rank-one (separable) functions, which are the product of two univaraiate functions, we show that the hyperbolic cross truncation is indeed the best if the functions have weak singularities on the domain or boundaries so that the spectral series has a finite order of power-law convergence. For functions smooth on the domain, and therefore blessed with exponentially convergent spectral series, we have failed to find any reasonable examples where the hyperbolic cross truncation is best.

A refined Poisson summation formula for certain Braverman-Kazhdan spaces

Braverman and Kazhdan (2000) introduced influential conjectures aimed at generalizing the Fourier transform and the Poisson summation formula. Their conjectures should imply that quite general Langlands L-functions have meromorphic continuations and functional equations as predicted by Langlands’ functoriality conjecture. As an evidence for their conjectures, Braverman and Kazhdan (2002) considered a setting related to the so-called doubling method in a later paper and proved the corresponding Poisson summation formula under restrictive assumptions on the functions involved. The connection between the two papers is made explicit in the work of Li (2018). In this paper, we consider a special case of the setting in Braverman and Kazhdan’s later paper and prove a refined Poisson summation formula that eliminates the restrictive assumptions of that paper. Along the way we provide analytic control on the Schwartz space we construct; this analytic control was conjectured to hold (in a slightly different setting) in the work of Braverman and Kazhdan (2002).

Windowed special affine Fourier transform

With the aim to circumvent the limitations of the special affine Fourier transform, we introduce a novel time–frequency transform namely the windowed special affine Fourier transform. We initiate our investigation by studying some fundamental properties of the proposed transform such as orthogonality relation, inversion formula and characterization of the range by employing the machinery of special affine Fourier transforms and operator theory. Continuing our endeavour, we propose a discrete analogue of the proposed windowed special affine Fourier transform and obtain the corresponding reconstruction formula. Besides, some potential applications of this new transform including windowed series expansion, Poisson summation formula, Paley–Wiener criterion and uncertainty principles are also given.

Sparse Nonuniform Frequency Sampling for Fast SFCW Radar Imaging

Stepped frequency continuous wave (SFCW) radar achieves wide bandwidth by synthesizing series of monochromatic pulses in a consecutive manner. Uniform frequency sampling is often performed with a constant frequency step, which in turn limits the maximum unambiguous range achievable by the radar to reliably distinguish targets. Thus, a small frequency step must be selected when clutters exist at long distances even though the target of interests is located at much closer distances. This is costly since a large number of frequencies must be synthesized, which leads to slow acquisition speed. In this paper, a sparse nonuniform frequency sampling method is proposed to effectively reduce the number of frequencies while suppressing aliasing effects from clutters. The Poisson Sum Formula is utilized to derive a deterministic formula for choosing a discrete set of frequencies within a specified frequency band. A corresponding frequency weighting formula is added in order to maintain the same target impulse response in time-domain as the one achieved by dense uniform frequency sampling. Numerical and experimental results are presented to demonstrate the improved performances of the proposed sparse sampling method for SFCW radar imaging.

A summation formula for the Rankin-Selberg monoid and a nonabelian trace formula

Let $F$ be a number field and let $\Bbb{A}_F$ be its ring of adeles. Let $B$ be a quaternion algebra over $F$ and let $\nu:B\to F$ be the reduced norm. Consider the reductive monoid $M$ over $F$ whose points in an $F$-algebra $R$ are given by $$ M(R):=\big\{\big(\gamma_1,\gamma_2\big)\in\big(B\otimes_F R\big)^2:\nu\big(\gamma_1\big)=\nu\big(\gamma_2\big)\big\}. $$ Motivated by an influential conjecture of Braverman and Kazhdan we prove a summation formula analogous to the Poisson summation formula for certain spaces of functions on the monoid. As an application, we define new zeta integrals for the Rankin-Selberg $L$-function and prove their basic properties. We also use the formula to prove a nonabelian twisted trace formula, that is, a trace formula whose spectral side is given in terms of automorphic representations of the unit group of $M$ that are isomorphic (up to a twist by a character) to their conjugates under a simple nonabelian Galois group.

The Wigner-Ville Distribution Associated with the Quaternion Offset Linear Canonical Transform

The Wigner-Ville distribution (WVD) and the quaternion offset linear canonical transform (QOLCT) are useful tools in signal analysis and image processing. The purpose of this paper is to define the Wigner-Ville distribution associated with the quaternionic offset linear canonical transform (WVD-QOLCT). Actually, this transform combines both the results and flexibility of the two transforms WVD and QOLCT. We derive some important properties of this transform such as inversion and Plancherel formulas, we establish a version of the Heisenberg inequality, Lieb’s theorem and we give the Poisson summation formula for the WVD-QOLCT.

Dual univariate interpolatory subdivision of every arity: Algebraic characterization and construction

A new class of univariate stationary interpolatory subdivision schemes of dual type is presented. As opposed to classical primal interpolatory schemes, these new schemes have masks with an even number of elements and are not step-wise interpolants. A complete algebraic characterization, which covers every arity, is given in terms of identities of trigonometric polynomials associated to the schemes. This characterization is based on a necessary condition for refinable functions to have prescribed values at the nodes of a uniform lattice, as a consequence of the Poisson summation formula. A strategy for the construction is then showed, alongside meaningful examples for applications that have comparable or even superior properties, in terms of regularity, length of the support and/or polynomial reproduction, with respect to the primal counterparts.

A new insight into the consistency of the SPH interpolation formula

In this paper, the consistency of the smoothed particle hydrodynamics (SPH) interpolation formula is investigated by analytical means. A novel error analysis is developed in n-dimensional space using the Poisson summation formula, which enables the simultaneous treatment of both the kernel and particle approximation errors for arbitrary particle distributions. New consistency integral relations are derived for the particle approximation, which correspond to the cosine Fourier transform of the kernel consistency conditions. The functional dependence of the error bounds on the SPH interpolation parameters, namely the smoothing length, h, and the number of particles within the kernel support, N, is demonstrated explicitly from which consistency conditions arise. As N→∞, the particle approximation converges to the kernel approximation independently of h provided that the particle mass scales with h as m∝hβ with β > n, where n is the spatial dimension. This implies that as h → 0, the joint limit m → 0, N→∞, and N → ∞ is necessary for complete convergence to the continuum, where N is the total number of particles. The analysis also reveals a dominant error term of the form (lnN)n/N for finite N, as it has long been conjectured based on the similarity between the SPH and the quasi-Monte Carlo estimates. When N≫1, the error of the SPH interpolant decays as N−1 independently of the dimension. This ensures approximate partition of unity of the kernel volume.

On spectral theory of the Riemann zeta function

Every nontrivial zero of the Riemann zeta function is associated as eigenvalue with an eigenfunction of the fundamental differential operator on a Hilbert-Polya space. It has geometric multiplicity one. A relation between nontrivial zeros of the zeta function and eigenvalues of the convolution operator is given. It is an analogue of the Selberg transform in Selbergs trace formula. Elements of the Hilbert-Polya space are characterized by the Poisson summation formula.