Hypercircle inequalities and sampling theory

Abstract

In this article the well-known hypercircle inequality is extended to the Riesz bases setting. A natural application for this new inequality is given by the estimation of the truncation error in nonorthogonal sampling formulas. Examples including the estimation of the truncation error for wavelet sampling expansions or for nonorthogonal sampling formulas in Paley–Wiener spaces are exhibited.

Keywords:

• Reproducing kernel Hilbert spaces
• Biorthonormal Riesz bases
• Hypercircle inequality
• Sampling formulas
• Multiresolution Analysis
• Frames
• AMS Subject Classifications: 94A20
• 44A05
• 46E22

Acknowledgments

The authors gratefully acknowledge the fruitful suggestions of Prof. Saburou Saitoh. This work has been supported by the grant BFM2000–0029 from the D.G.I. of the Spanish Ministerio de Ciencia y Tecnología.