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.
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.
Complete interpolating sequences for Paley-Wiener spaces and Muckenhoupt's ($A_p$) condition
Revista Matemática Iberoamericana, 2000
We describe the complete interpolating sequences for the Paley-Wiener spaces L p π (1 < p < ∞) in terms of Muckenhoupt's (A p ) condition. For p = 2, this description coincides with those given by ), Nikol'skii (1980 of the unconditional bases of complex exponentials in L 2 (−π, π). While the techniques of these authors are linked to the Hilbert space geometry of L 2 π , our method of proof is based on turning the problem into one about boundedness of the Hilbert transform in certain weighted L p spaces of functions and sequences.
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 of subspaces and applications to exponential bases in Sobolev spaces
2001
We give precise conditions under which the real interpolation space [Y_0,X_1]_s,p coincides with a closed subspace of the corresponding interpolation space [X_0,X_1]_s,p when Y_0 is a closed subspace of X_0 of codimension one. This result is applied to study the basis properties of nonharmonic Fourier series in Sobolev spaces H^s on an interval when 0<s<1. The main result: let E be a family of exponentials exp(i λ_n t) and E forms an unconditional basis in L^2 on an interval. 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 in [s_0,s_1] the exponential family is not an unconditional basis in its span.
Irregular stable sampling and interpolation in functional normed spaces
Boletim da Sociedade Paranaense de Matemática, 2022
We define the concepts of stable sampling set and stable interpolation set, uniqueness set and complete interpolation set for a normed space of functions. In addition we will show some relationships between these concepts. The main relationships arise when one wants to reduce an stable sampling set or to extend an stable interpolation set. We will prove that for Banach spaces verifying certain conditions, the complete interpolation sets are precisely the minimal stable sampling sets and are also the maximal stable interpolation sets. Finally we illustrate these results applying them to Paley-Wiener spaces, where we use a result by B. Matei, Yves Meyer and J. Ortega-Cerd´a based on the celebrated Fefferman theorem.
Paley–Wiener subspace of vectors in a Hilbert space with applications to integral transforms
Journal of Mathematical Analysis and Applications, 2009
The goal of this article is to introduce an analogue of the Paley-Wiener space of bandlimited functions, PW ω , in Hilbert spaces and then apply the general result to more specific examples. Guided by the role that the differentiation operator plays in some of the characterizations of the Paley-Wiener space, we construct a space of vectors using a selfadjoint operator D in a Hilbert space H, and denote this space by PW ω (D). The article can be virtually divided into two parts. In the first part we show that the space PW ω (D) has similar properties to those of the space PW ω , including an analogue of the Bernstein inequality and the Riesz interpolation formula. We also develop a new characterization of the abstract Paley-Wiener space in terms of solutions of Cauchy problems associated with abstract Schrödinger equations. Finally, we prove two sampling theorems for vectors in PW ω (D), one of which uses the notion of Hilbert frames and the other is based on the notion of variational splines in H. In the second part of the paper we apply our abstract results to integral transforms associated with singular Sturm-Liouville problems. In particular we obtain two new sampling formulas related to one-dimensional Schrödinger operators with bounded potential.
Interpolation and sampling sequences for mixed-norm spaces
arXiv (Cornell University), 2018
This paper extends the known characterization of interpolation and sampling sequences for Bergman spaces to the mixed-norm spaces. The Bergman spaces have conformal invariance properties not shared by the mixed-norm spaces. As a result, different techniques of proof were required. * Sections 1-3 of this paper are taken from the Ph.D. dissertation of the first author [11] under the direction of the second author. Section 4 on sampling was completed later.
Multidimensional Lagrange–Yen-Type Interpolation Via Kotel'nikov–Shannon Sampling Formulas
Ukrainian Mathematical Journal, 2003
Direct finite interpolation formulas are developed for the Paley-Wiener function spaces L 2 ♦ and L 2 [-π,π] d , where L 2 ♦ contains all bivariate entire functions whose Fourier spectrum is supported by the set ♦ = Cl{(u, v)||u| + |v| < π}, while in L 2 [-π,π] d the Fourier spectrum support set of its d-variate entire elements is [-π, π] d . The multidimensional Kotel'nikov-Shannon sampling formula remains valid when only finitely many sampling knots are deviated from the uniform spacing. By using this interpolation procedure, we truncate a sampling sum to its irregularly sampled part. Upper bounds of the truncation error are obtained in both cases. According to the Sun-Zhou extension of the Kadets 1 4 -theorem, the magnitudes of deviations are limited coordinatewise to 1 4 . To avoid this inconvenience, we introduce weighted Kotel'nikov-Shannon sampling sums. For L 2 [-π,π] d , Lagrange-type direct finite interpolation formulas are given. Finally, convergence-rate questions are discussed.
2011
Restricted non-linear approximation is a type of N-term approximation where a measure nu\nunu on the index set (rather than the counting measure) is used to control the number of terms in the approximation. We show that embeddings for restricted non-linear approximation spaces in terms of weighted Lorentz sequence spaces are equivalent to Jackson and Bernstein type inequalities, and also to the upper and lower Temlyakov property. As applications we obtain results for wavelet bases in Triebel-Lizorkin spaces by showing the Temlyakow property in this setting. Moreover, new interpolation results for Triebel-Lizorkin and Besov spaces are obtained.
Interpolation by Analytic Functions on Preduals of Lorentz Sequence Spaces
Glasgow Mathematical Journal, 2006
Let (e n) be the canonical basis of the predual of the Lorentz sequence space d * (w, 1). We consider the restriction operator R associated to the basis (e i) from some Banach space of analytic functions into the complex sequence space and we characterize the ranges of R.