Related papers: A Family of Unitary Operators Satisfying a Poisson…
The set of common numerical and analytical problems is introduced in the form of the generalized multidimensional discrete Poisson equation. It is shown that its solutions with square-summable discrete derivatives are unique up to a…
In this short note we show the equivalence of Fourier expansion and Poisson summation approaches for the series approximation of the exponential function $\exp ({-{t^2}/4})$. The application of the Poisson summation formula is shown to…
We derive special forms of the Poisson summation formula for even and odd functions, which are applied to obtain representations for Euler-type numbers and to sum various series related to elliptic functions.
We give a simple proof of the fact 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…
We prove a wave trace singularity formula for a family of generalised Laplacians defined by a Riemannian fibre bundle; for example, the superconnection curvature operator associated to the Bismut superconnection. It is explained how this…
Nature provides us with a restricted set of microscopic interactions. The question is whether we can synthesize out of these fundamental interactions an arbitrary unitary operator. In this paper we present a constructive algorithm for…
Basing on properties of the Mellin transform and Ramanujan's identities, which represent a ratio of products of Riemann's zeta- functions of different arguments in terms of the Dirichlet series of arithmetic functions, we obtain a number of…
We generalize the standard Poisson summation formula for lattices so that it operates on the level of theta series, allowing us to introduce noninteger dimension parameters (using the dimensionally continued Fourier transform). When…
We completely classify Fourier summation formulas, and in particular, all crystalline measures with quadratic decay. Our classification employs techniques from almost periodic functions, Hermite-Biehler functions, de Branges spaces and…
A unitary family is a family of unitary operators $U(x)$ acting on a finite dimensional hermitian vector space, depending analytically on a real parameter $x$. It is monotone if $\frac1i U'(x)U(x)^{-1}$ is a positive operator for each $x$.…
We characterize the solutions of the Poisson equation and the domain of its associated one-sided Hilbert transform for Ces\`aro bounded operators of fractional order. The results obtained fairly generalize the corresponding ones for…
We prove that a variety of oscillatory and polynomial Carleson operators are uniformly bounded on the family of parameters under considerations. As a particular application of our techniques, we prove uniform bounds for oscillatory Carleson…
We present an abstract result that characterizes the coincidence of certain classes of linear operators with the class of Cohen strongly summing linear operators. Our argument is extended to multilinear operators and, as a consequence, we…
We find asymptotic equalities for exact upper bounds of approximations by Fourier sums in uniform metric on classes of $2\pi$-periodic functions, representable in the form of convolutions of functions $\varphi$, which belong to unit balls…
In this paper, we study Hamiltonian operators which are sum of a first order operator and of a Poisson tensor, in two spatial independent variables. In particular, a complete classification of these operators is presented in two and three…
We present an improved version of commutator methods for unitary operators under a weak regularity condition. Once applied to a unitary operator, the method typically leads to the absence of singularly continuous spectrum and to the local…
In this note, we describe an interpretation of the (continuous) Fourier transform from the perspective of the Chinese Remainder Theorem. Some related issues, including a new derivation of Poisson summation formula, are discussed.
The Euler-Maclaurin summation formula is generalized to a modified form by expanding the periodic Bernoulli polynomials as its Fourier series and taking cuts, which includes both the Euler-Maclaurin summation formula and the Poission…
We give a natural definition of a Poisson Differential Algebra. Consistence conditions are formulated in geometrical terms. It is found that one can often locally put the Poisson structure on differential calculus in a simple canonical form…
We give several families of polynomials which are related by Eulerian summation operators. They satisfy interesting combinatorial properties like being integer-valued at integral points. This involves nearby-symmetries and a recursion for…