Entire functions and Müntz-Szász type approximation

Open Access
Publisher Website
Google Scholar
Abstract
Let be a bounded interval with . Under what conditions on the sequence of exponents can every function in or be approximated arbitrarily closely by linear combinations of powers ? What is the distance between and the closed span ? What is this closed span if not the whole space? Starting with the case of , C. H. Müntz and O. Szász considered the first two questions for the interval . L. Schwartz, J. A. Clarkson and P. Erdös, and the second author answered the third question for and also considered the interval . For the case of , L. Schwartz (and, earlier, in a limited way, T. Carleman) successfully used methods of complex and functional analysis, but until now the case of had proved resistant to a direct approach of that kind. In the present paper complex analysis is used to obtain a simple direct treatment for the case of . The crucial step is the construction of entire functions of exponential type which vanish at prescribed points not too close to the real axis and which, in a sense, are as small on both halves of the real axis as such functions can be. Under suitable conditions on the sequence of complex numbers , the construction leads readily to asymptotic lower bounds for the distances . These bounds are used to determine and to generalize a result for a boundary value problem for the heat equation obtained recently by V. J. Mizel and T. I. Seidman.

This publication has 13 references indexed in Scilit: