Entire functions and Müntz-Szász type approximation

Luxemburg, W. A. J.; Korevaar, J.

doi:10.1090/S0002-9947-1971-0281929-0

Entire functions and Müntz-Szász type approximation
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by W. A. J. Luxemburg and J. Korevaar PDF

Trans. Amer. Math. Soc. 157 (1971), 23-37 Request permission

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

Ralph Philip Boas Jr., Entire functions, Academic Press, Inc., New York, 1954. MR 0068627

Über die Approximation analytischer Funktionen durch lineare Aggregate von vorgegebenen Potenzen

17

M. L. Cartwright, Integral functions, Cambridge Tracts in Mathematics and Mathematical Physics, No. 44, Cambridge, at the University Press, 1956. MR 0077622
J. A. Clarkson and P. Erdös, Approximation by polynomials, Duke Math. J. 10 (1943), 5–11. MR 7813

A note on Fourier transforms

9

Børge Jessen and Aurel Wintner, Distribution functions and the Riemann zeta function, Trans. Amer. Math. Soc. 38 (1935), no. 1, 48–88. MR 1501802, DOI 10.1090/S0002-9947-1935-1501802-5
Tatsuo Kawata, On symmetric Bernoulli convolutions, Amer. J. Math. 62 (1940), 792–794. MR 2664, DOI 10.2307/2371489
J. Korevaar, A characterization of the sub-manifold of $C[a,b]$ spanned by the sequence $\{x_k{}^n\}$, Nederl. Akad. Wetensch., Proc. 50 (1947), 750–758 = Indagationes Math. 9, 360–368 (1947). MR 23368
B. Ja. Levin, Distribution of zeros of entire functions, American Mathematical Society, Providence, R.I., 1964. MR 0156975

On a class of non-vanishing functions

41

Norman Levinson, Gap and Density Theorems, American Mathematical Society Colloquium Publications, Vol. 26, American Mathematical Society, New York, 1940. MR 0003208
S. Mandelbrojt, Influence des propriétés arithmétiques des exposants dans une série de Dirichlet, Ann. Sci. Ecole Norm. Sup. (3) 71 (1954), 301–320 (French). MR 0068020
V. J. Mizel and T. I. Seidman, Observation and prediction for the heat equation, J. Math. Anal. Appl. 28 (1969), 303–312. MR 247301, DOI 10.1016/0022-247X(69)90029-8

Über den Approximationssatz von Weierstrass

Raymond E. A. C. Paley and Norbert Wiener, Fourier transforms in the complex domain, American Mathematical Society Colloquium Publications, vol. 19, American Mathematical Society, Providence, RI, 1987. Reprint of the 1934 original. MR 1451142, DOI 10.1090/coll/019
Laurent Schwartz, Étude des sommes d’exponentielles réelles, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 959, Hermann & Cie, Paris, 1943 (French). MR 0014502

Über die Approximation stetiger Funktionen durch lineare Aggregate von Potenzen

77

Aurel Wintner, On Analytic Convolutions of Bernoulli Distributions, Amer. J. Math. 56 (1934), no. 1-4, 659–663. MR 1507049, DOI 10.2307/2370961
Aurel Wintner, On symmetric Bernoulli convolutions, Bull. Amer. Math. Soc. 41 (1935), no. 2, 137–138. MR 1563036, DOI 10.1090/S0002-9904-1935-06035-5
Aurel Wintner, On Convergent Poisson Convolutions, Amer. J. Math. 57 (1935), no. 4, 827–838. MR 1507116, DOI 10.2307/2371018
Aurel Wintner, On a Class of Fourier Transforms, Amer. J. Math. 58 (1936), no. 1, 45–90. MR 1507134, DOI 10.2307/2371058