Sampling formula for convolution with a prolate

Abstract

Eigenfunctions of the Finite Fourier Transform, often referred to as ‘prolates’, are band-limited and highly concentrated at a finite time-interval. Both features are acquired by the convolution of a band-limited function with a prolate. This permits interpolation of such a convolution by the Walter and Shen sampling formula in terms of prolates, although the Fourier transform of the convolution is not necessarily even continuous and the concentration interval is twice as large as that of a prolate. Rigorous error estimates are given as dependent on the truncation limits. The accuracy achieved is tested by numerical examples.

Keywords:

• Sampling formula
• Band-limited function
• Time-concentrated function
• Finite Fourier transform
• Prolate spheroidal wave functions

Acknowledgements

This work is supported by the Royal Swedish Academy of Sciences, HPC2N/NSC/PDC, the Foundation of Strategic Research, project number 99232, and the Russian Foundation for Basic Research under Grant No. 05-01-00257. The authors thank the organizers of CMMSE 2006 for generous hospitality.