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Sampling and average sampling in quasi shift-invariant spaces. (English) Zbl 1458.94188

Summary: In this paper, we study the sampling and average sampling problem in a quasi shift-invariant space \(V_X(\varphi)\) where \(X\) is a discrete subset of \(\mathbb R\) and \(\varphi\) is a continuously differentiable positive definite function satisfying certain decay conditions. We show that any \(f\) belonging to \(V_X(\varphi)\) can be uniquely and stably reconstructed from its samples \(\{f(y_k) : k \in \mathbb Z\}\) as well as from its average samples provided sampling points \(\{y_k : k \in \mathbb Z\}\) are close enough. Further, iterative reconstruction algorithms for reconstruction of a function \(f\) belonging to \(V_X(\varphi)\) from its samples \(\{f(y_k) : k \in \mathbb Z\}\) as well as from its average samples are also provided.

MSC:

94A20 Sampling theory in information and communication theory
42C15 General harmonic expansions, frames
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