Related papers: Poisson summation formula and Box splines
A multidimensional generalization of Bailey's very-well-poised bilateral basic hypergeometric ${}_6\psi_6$ summation formula and its Dougall type ${}_5H_5$ hypergeometric degeneration for $q\to 1$ is studied. The multiple Bailey sum amounts…
A summation is a shift-invariant ${\rm R}$-module homomorphism from a submodule of ${\rm R}[[\sigma]]$ to ${\rm R}$ or another ring. [11] formalized a method for extending a summation to a larger domain by telescoping. In this paper, we…
We characterize the measures on R which have both their support and spectrum uniformly discrete. A similar result is obtained in R^n for positive measures.
The Macaulay2 package SumsOfSquares decomposes polynomials as sums of squares. It is based on methods to rationalize sum-of-squares decompositions due to Parrilo and Peyrl. The package features a data type for sums-of-squares polynomials,…
We prove a single sum formula for the linearization coefficients of the Bessel polynomials. In two special cases we show that our formula reduces indeed to Berg and Vignat's formulas in their proof of the positivity results about these…
In the first part of this paper, we present a direct proof of a Poisson summation formula on the Whittaker space of $\mathrm{GL}_2$, which underlies the local Hankel transform computed by H. Jacquet. In the second part, we derive a…
Using matrix inversion and determinant evaluation techniques we prove several summation and transformation formulas for terminating, balanced, very-well-poised, elliptic hypergeometric series.
We propose a deformed version of the generalized Heisenberg algebra by using techniques borrowed from the theory of pseudo-bosons. In particular, this analysis is relevant when non self-adjoint Hamiltonians are needed to describe a given…
We characterize the parameters $(\alpha,\beta,p,q)$ for which the condition $f|x|^\alpha\in L^p$ and $\widehat{f}|\xi|^\beta\in L^q$ implies the validity of the Poisson summation formula, thus completing the study of Kahane and…
We prove a previously conjectured closed form formula for the norm of the Jack polynomials in superspace with respect to a certain scalar product. The proof is mainly combinatorial and relies on the explicit expression in terms of…
We attempt to explain the ubiquity of tableaux and of Pieri and Cauchy formulae for combinatorially defined families of symmetric functions. We show that such formulae are to be expected from symmetric functions arising from representations…
We show that the Euler-MacLaurin formula for Riemann sums has an n-dimensional analogue in which intervals on the line get replaced by convex polytopes.
A factorization formula for certain automorphisms of a Poisson algebra associated to a quiver is proved, which involves framed versions of moduli spaces of quiver representations. This factorization formula is related to wall-crossing…
We study the notion of formal duality introduced by Cohn, Kumar, and Sch\"urmann in their computational study of energy-minimizing particle configurations in Euclidean space. In particular, using the Poisson summation formula we reformulate…
In this paper, we recast a special case of Mahler'c conjecture by the maximum value of box splines. This is the case of polytopes with at most $2n+2$ facets. An asymptotic formula for univariate box splines is given. Based on the formula,…
We give an overview of classical summation formulations, such as Poisson's and Voronoi's, and then turn to modern versions involving modular form coefficients. A new formula involving the coefficients of cusp forms on GL(3) is described,…
We present a decomposition principle for general regular Dirichlet forms satisfying a spatial local compactness condition. We use the decomposition principle to derive a Persson type theorem for the corresponding Dirichlet forms. In…
In this note, we present a simple summation formula for $k$-bonacci numbers. The derivation consists in obtaining the generating function of such numbers, and noting that its evaluation at a particular value yields a formula generalizing a…
In this paper we study the set of rational solutions of equations defined by power sums symmetric polynomials with coefficients in a finite field. We do this by means of applying a methodology which relies on the study of the geometry of…
In this note, we announce new regularity results for some locally integrable distributional solutions to Poisson's equation. This includes, for example, the standard solutions obtained by convolution with the fundamental solution. In…