On summability of eigenfunction expansions of piecewise smooth functions

Abstract

We consider two forms of eigenfunction expansions associated with an arbitrary elliptic differential operator with constant coefficients and order m, that is the multiple Fourier series and integrals. For the multiple Fourier integrals, we prove the convergence of the Riesz means of order s > (N − 3)/2 of piecewise smooth functions of N ≥ 2 variables. The same result is proved in the case of the N ≥ 3 dimensional multiple Fourier series.

Keywords:

• Fourier series
• Eigenfunction expansions
• Riesz means.

AMS 2000 Mathematics Subject Classifications: :

• Primary 42B08
• Secondary 42C14

Acknowledgements

This work was done within my visit to the Abdus Salam International Centre for Theoretical Physics, Trieste, Italy. I gratefully acknowledge the ICTP for the support and hospitality. I would also like to express my thanks to Applicable Analysis's reviewer for helpful remarks.