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Entire Functions and Muntz-Szasz Type Approximation

Luxemburg, W. A. J. and Korevaar, J. (1971) Entire Functions and Muntz-Szasz Type Approximation. Transactions of the American Mathematical Society, 157 (6). pp. 23-37. ISSN 0002-9947. doi:10.2307/1995828.

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Let [a, b] be a bounded interval with a>O. Under what conditions on the sequence of exponents {A,,} can every function in LP[a, b] or C[a, b] be approxi mated arbitrarily closely by linear combinations of powers xAn? What is the distance between xA and the closed span Sc(xAn)? What is this closed span if not the whole space? Starting with the case of L2, C. H. Muntz and 0. Szasz considered the first two questions for the interval [0, 1]. L. Schwartz, J. A. Clarkson and P. Erdos, and the second author answered the third question for [0, 1] and also considered the interval [a, b]. For the case of [0, 1], L. Schwartz (and, earlier, in a limited way, T. Carleman) successfully used methods of complex and functional analysis, but until now the case of [a, b] 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 [a, b]. 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 {An} the construction leads readily to asymptotic lower bounds for the distances dk=d{xAk, Sc(xAn, nAk)}. These bounds are used to determine Sc(xAn) and to generalize a result for a boundary value problem for the heat equation obtained recently by V. J. Mizel and T. I. Seidman.

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Additional Information:© 1971 American Mathematical Society. Work of first author supported in part by NSF grant GP-14133, that of the second author by NSF grant GP-8445.
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Issue or Number:6
Record Number:CaltechAUTHORS:20181009-141126373
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Official Citation:Luxemburg, W., & Korevaar, J. (1971). Entire Functions and Muntz-Szasz Type Approximation. Transactions of the American Mathematical Society, 157, 23-37. doi:10.2307/1995828
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:90205
Deposited By: Joy Painter
Deposited On:10 Oct 2018 17:28
Last Modified:16 Nov 2021 03:30

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