ON SUMMABILITY AND CONVERGENCE OF EIGENFUNCTION EXPANSIONS OF A DIFFERENTIAL OPERATOR

Abstract

Let be a positive elliptic operator with constant coefficients, and let be a region in . We consider the operator on , and we let be an extension of this operator with a positive lower bound. Let denote the spectral family of the operator . The operator or its Riesz mean will be considered on functions , , such that , where is a region with compact closure in . We will study the norm of the operator . We obtain definitive results when the point lies in one of the three regions: (l-1)\left(\frac{1}{p}-\frac{1}{2}\right) \right\}.$ SRC=http://ej.iop.org/images/0025-5734/20/2/A01/tex_sm_1868_img17.gif/> For , we construct an example of a function for which the Riesz mean of order of its spectral expansion diverges almost everywhere. For , we construct an analogous example for multiple Fourier series expansions.Bibliography: 26 items.