Function Spaces Inclusions and Rate of Convergence of Riemann-Type Sums in Numerical Integration

Abstract

In signal processing, discrete convolutions are usually involved in fast calculating coefficients of time-frequency decompositions like wavelet and Gabor frames. Depending on the regularity of the mother analyzing functions, one wants to detect the right resolution in order to achieve good approximations of coefficients. Local–global conditions on functions in order to get the convergence rate of Riemann-type sums to their scalar products in L ² are presented. Wiener amalgam spaces, in particular for the space-time domain and W(L ²,l ¹) for the frequency domain, give natural norms in order to estimate errors. In particular, relations between the rate of convergence of these series to integrals by increasing resolution and the (minimal) required Besov regularity are presented by means of functional and harmonic analysis techniques.

Keywords:

• Decomposition spaces
• Discrete convolutions
• Function spaces inclusions
• Sampling theory
• Wavelets
• Mathematics Subject Classifications 2000: 46S30
• 94A20
• 65T50
• 65T60

Acknowledgments

The author wants to thank Giuseppe De Marco (Department of Pure and Applied Mathematics, University of Padua, Italy), Hans G. Feichtinger (Department of Mathematics, University of Vienna, Austria), Walter Gautschi (Department of Computer Science, Purdue University, West Lafayette, Indiana) for the fruitful discussions, suggestions, and encouragement. The author has been supported by the Doctorate program in Computational Mathematics at the University of Padna, Italy.