Sampling in Shift-Invariant Spaces Generated by Hilbert Space-Valued Functions

IF 2.1 3区数学 Q1 MATHEMATICS, APPLIED

Mathematical Methods in the Applied Sciences Pub Date : 2025-01-15 DOI:10.1002/mma.10710

Md Hasan Ali Biswas, Rohan Joy, Felix Krahmer, Ramakrishnan Radha

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引用次数: 0

Abstract

In this paper, we investigate the problem of sampling and reconstruction in principal shift-invariant spaces generated by Hilbert space-valued functions. Given any signal $f$ and data point $x_{k} \in ℝ$ , the sample $f (x_{k})$ is stored along a sequence of directions ${ν_{k m}}_{m \in ℤ}$ . Specifically, the inner products $⟨ f (x_{k}), ν_{k m} ⟩$ are stored. First, we define what we mean by a stable set of sampling and provide equivalent conditions for proving that a given set is a stable set of sampling. We then present a reconstruction formula for $f$ from its integer samples $⟨ f (k), ν_{k m} ⟩$ . Finally, we address the cases of perturbed and irregular sampling, examining their impact on the reconstruction process.

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来源期刊

Mathematical Methods in the Applied Sciences 数学-应用数学

CiteScore

4.90

自引率

6.90%

发文量

798

审稿时长

6 months

期刊介绍： Mathematical Methods in the Applied Sciences publishes papers dealing with new mathematical methods for the consideration of linear and non-linear, direct and inverse problems for physical relevant processes over time- and space- varying media under certain initial, boundary, transition conditions etc. Papers dealing with biomathematical content, population dynamics and network problems are most welcome. Mathematical Methods in the Applied Sciences is an interdisciplinary journal: therefore, all manuscripts must be written to be accessible to a broad scientific but mathematically advanced audience. All papers must contain carefully written introduction and conclusion sections, which should include a clear exposition of the underlying scientific problem, a summary of the mathematical results and the tools used in deriving the results. Furthermore, the scientific importance of the manuscript and its conclusions should be made clear. Papers dealing with numerical processes or which contain only the application of well established methods will not be accepted. Because of the broad scope of the journal, authors should minimize the use of technical jargon from their subfield in order to increase the accessibility of their paper and appeal to a wider readership. If technical terms are necessary, authors should define them clearly so that the main ideas are understandable also to readers not working in the same subfield.