Sampling in the functional hilbert space induced by a hilbert space valued kernel

Abstract

This work focuses on the study of sampling formulas for the range space of a linear transform defined on a Hilbert space by means of a suitable kernel. The sampling property consists of the reconstruction of any function in through its values on an appropriate sequence of points by means of a sampling expansion involving these values. In our case, the sampling property is derived by assuming some requirements on the kernel of the linear transform. A converse result shows the generality of the required conditions in the Riesz bases setting. Finally, we deal with sampling formulas in the case where samples of a related function, the derivative for instance, are allowed in order to recover the initial function.

Keywords:

• Reproducing kernel Hilbert spaces
• Sampling formulas
• Biorthonormal Riesz bases
• AMS Subject Classifications: 94A20; 44A05; 46E22

Acknowledgments

The authors wish to thank the referee for her/his valuable and constructive comments. This work has been supported by the grant BFM2000–0029 from the D.G.I. of the Spanish Ministerio de Ciencia y Tecnología.