On essentially time and Hankel band-limited functions

Abstract

In this paper, we show that the circular prolate spheroidal wave functions (CPSWFs) are the most concentrate energy function on (0, T) among Hankel band-limited functions, here T is a positive real number. Hence, they best approximate each function in the set of essentially time- and Hankel band-limited signals than any other subspace of L ²(0,+∞). More precisely, using the theory of the CPSWFs, we show that the space spanned by the N first CPSWFs best approximate the set of essentially time- and Hankel band-limited signals than any other subspace of L ²(0,+∞) of the same dimension N.

Keywords:

• Hankel band-limited functions
• essentially time-limited function
• circular prolate spheroidal wave functions
• essentially time- and Hankel band-limited functions

Acknowledgements

The author would like to thank A. Karoui, for his advice and constructive discussions that have improved this paper. The author would also like to thank the anonymous referee for his careful reading of the manuscript, for his valuable suggestions and comments that have greatly improved the content of this work.