Wilson Bases and Modulation Spaces

We show that the recently discovered WILSON bases of exponential decay are unconditional bases for all modulation spaces on R, including the classical BESSEL potential spaces, the Segal algebra S_o, and the SCHWARTZ space. As a consequence we obtain new bases for spaces of entire functions. On the other hand, the WILSON bases are no unconditional bases for the ordinary L^p-spaces for p ≠ 2.     

A Riesz basis for Bargmann-Fock space related to sampling and interpolation

It is shown that the Bargmann-Fock spaces of entire functions, A^p (C),p≧1 have a bounded unconditional basis of Wilson type [DJJ] which is closely related to the reproducing kernel. From this is derived a new sampling and interpolation result for these spaces. Partially supported by grant AFOSR 90-0311.     

Sampling and Reconstruction in Distinct Subspaces Using Oblique Projections

Journal of Fourier Analysis and Applications - We study reconstruction operators on a Hilbert space that are exact on a given reconstruction subspace. Among those the reconstruction operator...     

Convergence Analysis of the Finite Section Method and Banach Algebras of Matrices

The finite section method is a classical scheme to approximate the solution of an infinite system of linear equations. Based on an axiomatic framework we present a convergence analysis of the finite section method for unstructured matrices on weighted ℓ ^p -spaces. In particular, the stability of the finite section method on ℓ ² implies its stability on weighted ℓ ^p -spaces. Our approach uses recent results from the theory of Banach algebras of matrices with off-diagonal decay. Furthermore, we demonstrate that Banach algebra theory provides a natural framework for deriving a finite section method that is applicable to large classes of unstructured non-hermitian matrices as well as to least squares problems.     

Sampling theorems on locally compact groups from oscillation estimates

We present a general approach to derive sampling theorems on locally compact groups from oscillation estimates. We focus on the L ²-stability of the sampling operator by using notions from frame theory. This approach yields particularly simple and transparent reconstruction procedures. We then apply these methods to the discretization of discrete series representations and to Paley–Wiener spaces on stratified Lie groups.     

Tight compactly supported wavelet frames of arbitrarily high smoothness

Based on Ron and Shen's new method for constructing tight wave-let frames, we show that one can construct, for any dilation matrix, and in any spatial dimension, tight wavelet frames generated by compactly supported functions with arbitrarily high smoothness.

     

Irregular sampling of wavelet and short-time Fourier transforms

We obtain irregular sampling theorems for the wavelet transform and the short-time Fourier transform. These sampling theorems yield irregular weighted frames for wavelets and Gabor functions with explicit estimates for the frame bounds.     

A Riesz basis for Bargmann-Fock space related to sampling and interpolation

It is shown that the Bargmann-Fock spaces of entire functions, A^p (C),p≧1 have a bounded unconditional basis of Wilson type [DJJ] which is closely related to the reproducing kernel. From this is derived a new sampling and interpolation result for these spaces.     

Convolution-Dominated Operators on Discrete Groups

We study infinite matrices A indexed by a discrete group G that are dominated by a convolution operator in the sense that for x ∈ G and some . This class of “convolution-dominated” matrices forms a Banach-*-algebra contained in the algebra of bounded operators on ℓ ²(G). Our main result shows that the inverse of a convolution-dominated matrix is again convolution-dominated, provided that G is amenable and rigidly symmetric. For abelian groups this result goes back to Gohberg, Baskakov, and others, for non-abelian groups completely different techniques are required, such as generalized L ¹-algebras and the symmetry of group algebras. K. G. was supported by the Marie-Curie Excellence Grant MEXT-CT 2004-517154.     

Banach Gabor frames with Hermite functions: polyanalytic spaces from the Heisenberg group

Gabor frames with Hermite functions are equivalent to sampling sequences in true Fock spaces of polyanalytic functions. In the L ²-case, such an equivalence follows from the unitarity of the polyanalytic Bargmann transform. We will introduce Banach spaces of polyanalytic functions and investigate the mapping properties of the polyanalytic Bargmann transform on modulation spaces. By applying the theory of coorbit spaces and localized frames to the Fock representation of the Heisenberg group, we derive explicit polyanalytic sampling theorems which can be seen as a polyanalytic version of the lattice sampling theorem discussed by J.M. Whittaker in Chapter 5 of his book Interpolatory Function Theory.