Entire functions and Müntz-Szász type approximation
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- by W. A. J. Luxemburg and J. Korevaar
- Trans. Amer. Math. Soc. 157 (1971), 23-37
- DOI: https://doi.org/10.1090/S0002-9947-1971-0281929-0
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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.References
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Bibliographic Information
- © Copyright 1971 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 157 (1971), 23-37
- MSC: Primary 30.70; Secondary 41.00
- DOI: https://doi.org/10.1090/S0002-9947-1971-0281929-0
- MathSciNet review: 0281929