Poisson Summation Research Articles

The Casimir effect for parallel plates revisited

The Casimir effect for a massless scalar field with Dirichlet and periodic boundary conditions (bc’s) on infinite parallel plates is revisited in the local quantum field theory (lqft) framework introduced by Kay [Phys. Rev. D 20, 3052 (1979)]. The model displays a number of more realistic features than the ones he treated. In addition to local observables, as the energy density, we propose to consider intensive variables, such as the energy per unit area ε, as fundamental observables. Adopting this view, lqft rejects Dirichlet (the same result may be proved for Neumann or mixed) bc, and accepts periodic bc: in the former case ε diverges, in the latter it is finite, as is shown by an expression for the local energy density obtained from lqft through the use of the Poisson summation formula. Another way to see this uses methods from the Euler summation formula: in the proof of regularization independence of the energy per unit area, a regularization-dependent surface term arises upon use of Dirichlet bc, but not periodic bc. For the conformally invariant scalar quantum field, this surface term is absent due to the condition of zero trace of the energy momentum tensor, as remarked by De Witt [Phys. Rep. 19, 295 (1975)]. The latter property does not hold in the application to the dark energy problem in cosmology, in which we argue that periodic bc might play a distinguished role.

The sampling theorem and coherent state systems

The Poisson Summation Formula and the sampling theorem for band limited signals, well known in the context of Fourier transformation theory, are analyzed from the perspective of the Zak basis and coherent state systems associated with the Heisenberg-Weyl group. In particular, we rephrase the content of the sampling theorem in terms coherent states and show that this in turn permits extensions, which allow us to make specific statements concerning standard and generalized coherent state systems on von Neumann or finer lattices.

A planar dipole array formed by small resonant particles

AbstractIn this article the complex interaction constant of an infinite planar dipole array is presented. The imaginary part can be expressed in a closed form and the real part can be formulated in a rapidly convergencing manner using the Poisson summation and the Fourier transformation. At a certain frequencies a resonator can be realized loading the dipole inclusions. Inductive, series resonance, and parallel resonance electric circuit loads are considered. © 2007 Wiley Periodicals, Inc. Microwave Opt Technol Lett 49: 2419–2422, 2007; Published online in Wiley InterScience (www.interscience.wiley.com) DOI 10.1002/mop.22730

Time Domain Weyl's Identity and the Causality Trick Based Formulation of the Time Domain Periodic Green's Function

Time domain Floquet modes for analyzing scattering (radiation) from periodically arranged planar scatterers (sources) have been presented recently in a series of papers by Felsen and Capolino. In these papers, they have investigated phenomenological details of transient scattering (radiation) by such structures. Typically, time domain Floquet modes are derived either by Fourier transforming their frequency domain counterparts or using spatial synthesis using Poisson summation. This paper proposes an alternative means to develop similar expressions and is motivated by the possibility of easily extending the method to the analysis of periodic structures in a layered medium. As will be shown, the time domain Weyl's identity can be used to construct the periodic Green's function using both homogeneous and inhomogeneous plane waves. We will also derive expressions using the Causality Trick. This technique involves using time-symmetric (non-causal) signals; however, it yields much simpler expressions that involve only homogeneous time domain plane waves and the results are identical to those obtained using the time domain Weyl's identity for the time region of interest. Finally, we will use the Whittaker formulation, which relies on real signals, to derive time domain Floquet modes. This formulation is applicable for simultaneous excitation only.

A simple general proof of the Krazer–Prym theorem and related famous formulae resolving convergence properties of Coulomb series in crystals

The Krazer–Prym theorem extending the Poisson summation formula to oblique lattices is proved as a direct consequence of translational symmetry. A generalized Ewald approach based on this theorem is reproduced. The two other famous approaches proposed by Nijboer and De Wette, and by Harris and Monkhorst, are derived in the same fashion. The absence of any uniform contribution to bulk electrostatic potentials in each of those treatments is emphasized as a property of locally neutral systems subject to translational symmetry. It is found that an analytic evaluation of the Ewald variable parameter naturally follows from the treatment of the Coulomb lattice characteristic in terms of the Krazer–Prym theorem. The extension of those characteristics to off-lattice points is also discussed.

An accurate closed-form expression for the partition function of Morse oscillators

On the basis of the Poisson summation formula, an explicit expression is proposed to evaluate the vibrational partition function for a mode with either negative or positive anharmonicity. This formula gives numerical values almost identical to exact values over the entire temperature range from zero to infinity. The developed approximation will also be available in numerical calculations of the whole vibrational partition function for polyatomic molecules.

Systematic investigation of two-dimensional static array sums

We discuss general properties of doubly periodic sums over the square lattice, linking phased, Bloch-type sums in the direct lattice with displaced sums in the reciprocal lattice using the Poisson summation formula. We discuss cardinal points, where the sums reduce to a single product of Dirichlet L functions, and exhibit all cardinal points for the square lattice. We introduce a new method for evaluating lattice sums and illustrate this by solving low order systems of sums of integer order 2,3,4,5. For the last case, the analytic expressions for the sums involve complementary L functions, or alternatively L functions with complex characters.

On a question of Igusa III: A generalized Poisson formula for pairs of polynomials

This paper is the first response to a question of Igusa that asked for a multivariable extension of his adelic Poisson formula in one variable. We prove a Poisson summation formula over the adèles between two distributions, depending upon two variables. Each distribution is defined by a mapping P : ℚ n → ℚ 2 whose components are homogeneous polynomials, and whose critical locus is a union of lines.

Radiation from Fibonacci-type Quasiperiodic Arrays on Dielectric Substrates

We present a simple prototype study of electromagnetic radiation by a one-dimensional quasiperiodic Fibonacci-type array laid on a grounded dielectric slab, extending our previous free-space studies. Analytic parameterization of the interaction between aperiodic-order-induced "quasi-Floquet" waves and slab-induced surface/leaky-waves is addressed, for infinite and truncated arrays, via generalized Poisson summation and uniform asymptotics. Accuracy and computational effectiveness of the proposed parameterizations are assessed via numerical comparisons against an independently-generated reference solution (element-by-element synthesis).

The sampling theorem and coherent state systems in quantum mechanics

The well-known Poisson Summation Formula is analysed from the perspective of the coherent state systems associated with the Heisenberg–Weyl group. In particular, it is shown that the Poisson Summation Formula may be viewed abstractly as a relation between two sets of bases (Zak bases) arising as simultaneous eigenvectors of two commuting unitary operators in which geometric phase plays a key role. The Zak bases are shown to be interpretable as generalized coherent state systems of the Heisenberg–Weyl group and this, in turn, prompts analysis of the sampling theorem (an important and useful consequence of the Poisson Summation Formula) and its extension from a coherent state point of view leading to interesting results on the properties of von Neumann and finer lattices based on standard and generalized coherent state systems.

Lp([0,1])-characterizations of multi-knot piecewise linear spectral sequences*

Abstract There exists a class of new orthonormal basis for L2([0,1]), whose exponential parts are multi-knot piecewise linear functions called spectral sequences. In this paper, we show that these bases constitute bases, but not unconditional bases, for L p ([0,1])with 1 < p < ∞, p≠2. In addition, we give the corresponding convergence theorem in L p , Carleson-Hunt theorem on almost everywhere convergence, Littlewood-Paley theorem and Poisson summation formula related to these bases.. * Supported by National Natural Science Foundation of China (Grant No. 10371122) ad the President Foundation of Graduate School of the Chinese Academy of Sciences (Grant No. yzjj200505)

A unified framework for the Sussman, Moyal, and Janssen formulas

This paper derives three fundamental identities in the radar and sonar literature, namely, Sussman' identity for ambiguity functions, Moyal's formula which establishes the value of the inner product between two scattering functions, and Janseen's formula which establishes identities for mixed inner products between waveforms and Gabor wavelets. Starting from the fundamental convolution identity, we derive Sussman's identity. Following from an initial value theorem of Fourier analysis, we obtained Moyal's formula. Following from Poisson's sum formula and an initial value theorem, we also obtained Janssen's equality. The relationship between these three identities is as follows: Janssen's formula is a sampled-data version of Moyal's formula, and both follow from Sussman's identity. In turn, Sussman's identity is a consequence of the fundamental convolution identity.

Schlömilch series that arise in diffraction theory and their efficient computation

We are concerned with a certain class of Schlomilch series that arise naturally in the study of diffraction problems when the scatterer is a semi-infinite periodic structure. By combining new results derived from integral representations and the Poisson summation formula with known identities, we obtain expressions which enable the series to be computed accurately and efficiently. Many of the technical details of the derivations are omitted; they can, however, be obtained from Linton 2005 Schlomilch series that arise in diffraction theory and their efficient computation Technical Report Loughborough University available online at http://www-staff.lboro.ac.uk/~macml1/schlomilch-techreport.pdf.

La formule de Plancherel pour les groupes de Lie presque algébriques réels

We give a proof of the Plancherel formula for real almost algebraic groups in the philosophy of the orbit method, following the lines of the one given by M. Duflo and M. Vergne for simply connected semisimple Lie groups. Main ingredients are: (1) Harish-Chandra's descent method which, interpreting Plancherel formula as an equality of semi-invariant generalized functions, allows one to reduce it to a neighbourhood of zero in the Lie algebra of the centralizer of any elliptic element; (2) character formula for representations constructed by M. Duflo, we recently proved; (3) Poisson–Plancherel formula near elliptic elements s in good position, a generalization of the classical Poisson summation formula expressing the Fourier transform of the sum of a series of Harish-Chandra type elliptic orbital integrals in the Lie algebra centralizing s as a generalized function supported on a set of admissible regular forms in the dual of this Lie algebra.

Distribution of the Fractional Parts of Random Vectors: The Gaussian Case. II

This paper considers a fractional distribution of an s-dimensional Gaussian random vector. Inequalities for the distribution deviation from the uniform distribution are proved. The proofs use the Poisson summation formula and some facts from the theory of representation of integers by square forms. The main attention of this part of the paper is devoted to the case of small values of s. The case of large values of s will be consider additionally.

PERIODIZATION OF FRACTIONAL INTEGRALS

The periodization of functions is well known in harmonic analysis (in particular, in application to sampling of signals), its central point being the Poisson summation formula.The notion of periodization for fractional integration appears in the paper H. Weyl (1917) well known to experts in Fractional Calculus. The periodic fractional integral (Weyl integral) of a 2π-periodic function f(x), generally speaking, coincides with the properly interpreted Liouville fractional integral of f. In fact, this is nothing else but the statement that the periodic Weyl fractional kernel is the periodization of the Liouville kernel.We present a general approach to the periodization of functions of one variable, but the main focus is made on the periodization of fractional integrals of functions of two variables. There exist many forms of multidimensional fractional integro-differentiation. We dwell on the case of the Riesz fractional integrals (Riesz potentials) of two variables and show that already in this case we come across different effects, depending on whether we use the repeated periodization, first in one variable, and afterwards in another one, or the so called double periodization. We show that the naturally introduced doubly-periodic Weyl-Riesz kernel of order 0 <α 2 in general coincides with the periodization of the Riesz kernel, the repeated periodization being possible for all 0 <α < 2, while the double one is applicable only for 0 <α 1. This is obtained as a realization of a certain general scheme of periodization (repeated or double periodization).

An efficient approach for the analysis of irregularly contoured planar phased arrays with tapered excitation in complex environments

An efficient uniform high‐frequency representation of the radiation from a large irregularly contoured planar tapered phased array, which is accurate throughout the near, intermediate, and far field zones, is presented. The planar array is regarded as the superposition of a sequence of parallel line arrays, and each one of them is represented in terms of equivalent continuous tapered lines via the Poisson summation formula. The amplitude tapering along the line is asymptotically approximated by a physically based, observation‐dependent linear combination of a small number (seven at most) of equiamplitude linearly phased excitations, which allows the reconstruction of the dominant contributions. After superimposing the asymptotic contributions of each line array, the total radiated field from the planar array is described in terms of rays emerging from the actual endpoints on the array rim and from a discrete sequence of points in linear loci on the surface of the array. As a consequence, high‐frequency techniques may effectively be used to describe the interaction of these rays with the surrounding environment.

Field patterns of resonant noncircular closed-loop arrays: further analysis

Previous papers have introduced the continuous-current model for resonant closed-loop arrays of cylindrical dipoles and, for a certain class of array shapes, have determined the resulting field patterns analytically. Those analytical calculations involve several approximations. In the present paper, we first study the aforementioned model numerically for the case of elliptical closed-loop arrays. The results-which involve fewer approximations than those of the previous papers, but which refer to different array shapes-imply that elliptical arrays have a bidirectional, highly directive field and, unlike conventional arrays, have no sidelobes. Combining analytical and numerical methods, we then investigate the relationship between the aforementioned continuous-current model and what we call the discrete-current model which, in a certain sense, is more realistic. The results indicate that, subject to suitable conditions, the two models yield virtually identical results. In our analytical investigations, an important tool is the Poisson summation formula for finite sums; we have devoted a stand-alone section to this formula which supplements recent, unrelated publications in antenna theory.

Image solution for clamped finite beams

An image solution for the forced response of a clamped finite beam is developed. The finite beam is replaced by an infinite beam under spatially periodic excitation and periodic support reactions. The response is then found by a Fourier transform approach, making use of the periodicity. An explicit expression is found, using the Poisson sum formula. The aim is to describe the potential of using the image method for clamped structural acoustic and vibration problems.

SAMPLING AND OVERSAMPLING IN SHIFT-INVARIANT AND MULTIRESOLUTION SPACES I: VALIDATION OF SAMPLING SCHEMES

We ask what conditions can be placed on generators φ of principal shift invariant spaces to ensure the validity of analogues of the classical sampling theorem for bandlimited signals. Critical rate sampling schemes lead to expansion formulas in terms of samples, while oversampling schemes can lead to expansions in which function values depend only on nearby samples. The basic techniques for validating such schemes are built on the Zak transform and the Poisson summation formula. Validation conditions are phrased in terms of orthogonality, smoothness, and self-similarity, as well as bandlimitedness or compact support of the generator. Effective sampling rates which depend on the length of support of the generator or its Fourier transform are derived.