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Characterization of local sampling sequences for spline subspaces

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Abstract

The sampling theorem is one of the most powerful tools in signal analysis. It says that to recover a function in certain function spaces, it suffices to know the values of the function on a sequence of points. Most of known results, e.g., regular and irregular sampling theorems for band-limited functions, concern global sampling. That is, to recover a function at a point or on an interval, we have to know all the samples which are usually infinitely many. On the other hand, local sampling, which invokes only finite samples to reconstruct a function on a bounded interval, is practically useful since we need only to consider a function on a bounded interval in many cases and computers can process only finite samples. In this paper, we give a characterization of local sampling sequences for spline subspaces, which is equivalent to the celebrated Schönberg-Whitney Theorem and is easy to verify. As applications, we give several local sampling theorems on spline subspaces, which generalize and improve some known results.

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Authors and Affiliations

  1. Department of Mathematics and LPMC, Nankai University, Tianjin, 300071, China

    Wenchang Sun & Xingwei Zhou

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  1. Wenchang Sun
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  2. Xingwei Zhou
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Correspondence to Wenchang Sun.

Additional information

Communicated by Qiyu Sun.

This work was supported partially by the National Natural Science Foundation of China (10571089 and 60472042), the Program for New Century Excellent Talents in Universities, and the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

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Sun, W., Zhou, X. Characterization of local sampling sequences for spline subspaces. Adv Comput Math 30, 153–175 (2009). https://doi.org/10.1007/s10444-008-9062-y

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  • DOI: https://doi.org/10.1007/s10444-008-9062-y

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