Sampling of Paley-Wiener functions on stratified groups

We consider a generalization of entire functions of spherical exponential type on stratified groups. An analog of the Paley-Wiener theorem is given. We also show that every spectral entire function on a stratified group is uniquely determined by its values on some discrete subgroups. The main result of the article is reconstruction formula of spectral entire functions from their values on discrete subgroups using Lagrangian splines.     

Mean oscillation of functions and the Paley-Wiener space

The oscillatory behavior of functions with compactly supported Fourier transform is characterized in a quantified way using various function spaces. In particular, the results in this article show that the oscillations of a function at large scale are comparable to the oscillations of its samples on an appropriate discrete set of points. Several open questions about spaces of sequences are answered and applications in the study of commutator operators on the Paley-Wiener space are shown. Acknowledgements and Notes. Supported in part by NSF grants DMS 9303363 and DMS 9623251.     

Real Paley-Wiener Theorems

The aim of this paper is to exhibit a real Paley–Wienerspace sitting inside the Schwartz space, and to give a quickand simple proof of a Paley–Wiener-type theorem. A simpleand elementary proof of a theorem postulated by H. H. Bang isalso given. 2000 Mathematics Subject Classification 42A38.     

A spectral Paley-Wiener theorem

The Fourier inversion formula in polar form is \(f(x) = \int_0^\infty {P_\lambda } f(x)d\lambda \) for suitable functionsf on ? ⁿ , whereP _λ f(x) is given by convolution off with a multiple of the usual spherical function associated with the Euclidean motion group. In this form, Fourier inversion is essentially a statement of the spectral theorem for the Laplacian and the key question is: how are the properties off andP _λ f related? This paper provides a Paley-Wiener theorem within this avenue of thought generalizing a result due to Strichartz and provides a spectral reformulation of a Paley-Wiener theorem for the Fourier transform due to Helgason. As an application we prove support theorems for certain functions of the Laplacian.     

Paley-Wiener type theorems by transmutations

The classical Paley-Wiener theorem for functions in L _dx ² relates the growth of the Fourier transform over the complex plane to the support of the function. In this work we obtain Paley-Wiener type theorems where the Fourier transform is replaced by transforms associated with self-adjoint operators on L _dμ ² , with simple spectrum, where dμ is a Lebesgue-Stieltjes measure. This is achieved via the use of support preserving transmutations. Communicated by Paul L. Butzer     

Interpolation in Bernstein and Paley-Wiener spaces

We consider interpolation of discrete functions by continuous ones with restriction on the size of spectra. We discuss a sharp contrast between the cases of compact and unbounded spectra. In particular we construct ‘universal’ spectra of small measure which deliver positive solution of the interpolation problem in Bernstein spaces for every discrete sequence of knots.     

Sampling in Paley-Wiener spaces on combinatorial graphs

A notion of Paley-Wiener spaces on combinatorial graphs is introduced. It is shown that functions from some of these spaces are uniquely determined by their values on some sets of vertices which are called the uniqueness sets. Such uniqueness sets are described in terms of Poincare-Wirtinger-type inequalities. A reconstruction algorithm of Paley-Wiener functions from uniqueness sets which uses the idea of frames in Hilbert spaces is developed. Special consideration is given to the -dimensional lattice, homogeneous trees, and eigenvalue and eigenfunction problems on finite graphs.

     

Toeplitz and Hankel operators on the Paley-Wiener space

The basic theory of Toeplitz and Hankel operators acting on the Paley-Weiner space is developed. This includes criteria for boundedness, compactness, being of finite rank, and membership in the Schatten-von Neumann ideals. Similar questions are considered for the related operators formed by commuting the discrete Hilbert transform with a multiplication operator.Supported in part by a grant from the National Science Foundation.     

Weighted interpolation in Paley-Wiener spaces and finite-time controllability

This paper considers the solution of weighted interpolation problems in model subspaces of the Hardy space H² that are canonically isometric to Paley-Wiener spaces of analytic functions. A new necessary and sufficient condition is given on the set of interpolation points which guarantees that a solution in H² can be transferred to a solution in a model space. The techniques used rely on the reproducing kernel thesis for Hankel operators, which is given here with an explicit constant. One of the applications of this work is to the finite-time controllability of diagonal systems specified by a C₀ semigroup.