Sampling and Interpolation Problems for Vector Valued Signals in the Paley–Wiener Spaces (original) (raw)

Sampling and Interpolation Problems for Vector Valued Signals in the Paley–Wiener Spaces

IEEE Transactions on Signal Processing, 2000

An approach to solving sampling and interpolation problems in the case of non-separated sequences is developed. The approach is based on the duality between Riesz bases of exponential divided differences and group sampling and interpolating sequences. The complete description of group sampling and interpolating sequences for the Paley-Wiener spaces is obtained and stability properties of these sequences are discussed.

Construction of Sampling and Interpolating Sequences for Multi-Band Signals. The Two-Band Case

International Journal of Applied Mathematics and Computer Science, 2007

Recently several papers have related the production of sampling and interpolating sequences for multi-band signals to the solution of certain kinds of Wiener-Hopf equations. Our approach is based on connections between exponential Riesz bases and the controllability of distributed parameter systems. For the case of two-band signals we derive an operator whose invertibility is equivalent to the existence of a sampling and interpolating sequence, and prove the invertibility of this operator.

Interpolation of functions from generalized Paley–Wiener spaces

Journal of Approximation Theory, 2005

We consider the problem of reconstruction of functions f from generalized Paley-Wiener spaces in terms of their values on complete interpolating sequence {z n }. We characterize the set of data sequences {f (z n)} and exhibit an explicit solution to the problem. Our development involves the solution of a particular* problem.

Exponential bases, Paley–Wiener spaces and applications

Journal of Functional Analysis, 2015

We investigate the connection between translation bases for Paley-Wiener spaces and exponential Fourier bases for a domain. We apply these results to the characterization of vector-valued time-frequency translates of a Paley-Wiener "window" signal.

Riesz bases in related to sampling in shift-invariant spaces

2005

The Fourier duality is an elegant technique to obtain sampling formulas in Paley-Wiener spaces. In this paper it is proved that there exists an analogous of the Fourier duality technique in the setting of shift-invariant spaces. In fact, any shiftinvariant space V ϕ with a stable generator ϕ is the range space of a bounded oneto-one linear operator T between L 2 (0, 1) and L 2 (R). Thus, regular and irregular sampling formulas in V ϕ are obtained by transforming, via T , expansions in L 2 (0, 1) with respect to some appropriate Riesz bases.

Sampling, interpolation and Riesz bases in small Fock spaces

We give a complete description of Riesz bases of reproducing kernels in small Fock spaces. This characterization is in the spirit of the well known Kadets-Ingham 1/4 theorem for Paley-Wiener spaces. Contrarily to the situation in Paley-Wiener spaces, a link can be established between Riesz bases in the Hilbert case and corresponding complete interpolating sequences in small Fock spaces with associated uniform norm. These results allow to show that if a sequence has a density stricly different from the critical one then either it can be completed or reduced to a complete interpolating sequence. In particular, this allows to give necessary and sufficient conditions for interpolation or sampling in terms of densities.

Interpolation by weighted Paley–Wiener spaces associated with the Dunkl transform

Journal of Mathematical Analysis and Applications, 2012

Given α > − 1 2 , σ > 0 and 1 p < ∞, we study the interpolation problem in the space PW p,α σ of entire functions f : C → C of exponential type σ for which R | f (x)| p × |x| 2α+1 dx < ∞, with nodes of interpolation at s j /σ , j ∈ Z, where {s j : j ∈ N} is the increasing sequence of all positive roots of the Bessel function J α+1 (z) of order α + 1, and s j = −s − j for all j ∈ Z. We prove that if 4(α+1) 2α+3 := p 1 (α) < p < p 2 (α) := 4(α+1) 2α+1 , the interpolation problem f ∈ PW p,α σ and f σ −1 s j = c j for all j ∈ Z has a unique solution for every sequence {c j } of complex numbers satisfying j∈Z |c j | p (1+ | j|) 2α+1 < ∞, and that if p p 1 (α), the corresponding interpolation problem may not have a solution, and that the solution, if exists, is unique if and only if p p 2 (α). Finally, we show that R f (x) p |x| 2α+1 dx ∼ σ −2α−2 j∈Z f s j σ −1 p 1 + | j| 2α+1 , with the constant of equivalence depending only on p and α, holds for all entire functions f of exponential type σ if and only if p 1 (α) < p < p 2 (α).

Interpolation of subspaces and applications to exponential bases

2000

We give precise conditions under which the real interpolation space [Y 0 , X 1 ] θ,p coincides with a closed subspace of [X 0 , X 1 ] θ,p when Y 0 is a closed subspace of codimension one. We then apply this result to nonharmonic Fourier series in Sobolev spaces H s (−π, π) when 0 < s < 1. The main result: let E be a family of exponentials exp(iλ n t) and E forms an unconditional basis in L 2 (−π, π). Then there exist two number s 0 , s 1 such that E forms an unconditional basis in H s for s < s 0 , E forms an unconditional basis in its span with codimension 1 in H s for s 1 < s. For s 0 ≤ s ≤ s 1 the exponential family is not an unconditional basis in its span.