On summability of eigenfunction expansions of piecewise smooth functions
by
Ravshan R. Ashurov
Lead Scientific Researcher at the Institute of Mathematics and Information Technologies, Uzbek Academy of Sciences

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.

We prove the following theorems:

Theorem 1. Let N ≥ 3 and the set L be convex. Then for every piecewise smooth function f with the surface of discontinuity G the Riesz means E_l^s f(x) of order s=(N-3)/2 are uniformly bounded on each compact set K ⊂ R^N \G. If s > (N-3)/2 then

lim l→+∞  (2p)-N ó õ A(x) < l  æ è 1- A(x) l ö ø s   ^ f   (x)eixxdx = f(x)

uniformly on x ∈ K ⊂ R^N \G.

Theorem 2. (i) Let N=2 and A(D) be an arbitrary homogeneous elliptic differential operator. Then for any piecewise smooth function f with the surface of discontinuity G the Riesz means E_l^s f(x) of order s > -1/2, defined as in (1), uniformly converge on x ∈ K ⊂ R² G.

(ii) If s=-1/2 then the corresponding Riesz means are uniformly bounded on each compact set K ⊂ R² \G.

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Date received: February 6, 2008

Copyright © 2008 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cavq-33.