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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Entire functions and Müntz-Szász type approximation

Authors: W. A. J. Luxemburg and J. Korevaar
Journal: Trans. Amer. Math. Soc. 157 (1971), 23-37
MSC: Primary 30.70; Secondary 41.00
MathSciNet review: 0281929
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Abstract: Let $ [a,b]$ be a bounded interval with $ a \geqq 0$. Under what conditions on the sequence of exponents $ \{ {\lambda _n}\} $ can every function in $ {L^p}[a,b]$ or $ C[a,b]$ be approximated arbitrarily closely by linear combinations of powers $ {x^\lambda }n$? What is the distance between $ {x^\lambda }$ and the closed span $ {S_c}({x^\lambda }n)$? What is this closed span if not the whole space? Starting with the case of $ {L^2}$, C. H. Müntz and O. Szász considered the first two questions for the interval $ [0, 1]$. L. Schwartz, J. A. Clarkson and P. Erdös, 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 $ \{ {\lambda _n}\} $, the construction leads readily to asymptotic lower bounds for the distances $ {d_k} = d\{ {x^{{\lambda _k}}},{S_c}({x^{{\lambda _n}}},n \ne k)\} $. These bounds are used to determine $ {S_c}({x^{{\lambda _n}}})$ 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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Keywords: Müntz-Szász type approximation, approximation by powers $ {x^{{\lambda _n}}}$, span of powers $ {x^{{\lambda _n}}}$, closed span, approximation in $ {L^p}[a,b]$, approximation in $ C[a,b]$, asymptotic inequalities for distances, entire functions of exponential type, zeros, functions of exponential type, separation condition for zeros, infinite products, smallness, finite Fourier transforms, Fourier inversion, Paley-Wiener theorem, supporting interval, harmonic function, subharmonic function, Poisson integral, maximum principle, boundary value problems, heat equation
Article copyright: © Copyright 1971 American Mathematical Society

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