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On a problem of G. Freud. (English) Zbl 0561.41019
A classical result of S. N. Bernstein states that if f(x) is a continuous function on [-1,1] and \(E_ n(f)=\inf \{\| f-P\|_{\infty}: P\) is a polynomial of degree at most \(n\}\), then \(\limsup_{n\to \infty}| E_ n(f)|^{1/n}=1/p<1\) if and only if f can be extended to an analytic function u(z) in the interior of an ellipse with foci at \(\pm 1\) and with the sum of the half-axes equal to p. The author considers a variation of this result, where the extension \(u(z)=u(x,t)\) of the function f is desired to satisfy the heat equation \(\partial^ 2u/\partial x^ 2=\partial u/\partial t\), \(-\infty <x<\infty\), \(t<0\). Some results similar to the Bernstein result are obtained by requiring that the starting function f be in the class \(L^ 2(-\infty,\infty)\) (rather than a continuous function) and using entire functions of exponential type at most \(\sigma\) (for some fixed \(\sigma)\) as the approximating functions (in place of polynomials of degree at most n), where the approximation is relative to the \(L^ 2\) norm.
Reviewer: P.Lappan
41A30 Approximation by other special function classes
30E05 Moment problems and interpolation problems in the complex plane
30B40 Analytic continuation of functions of one complex variable
Full Text: DOI
[1] Bernstein, S.N, Leçons sur LES propriétés extrémales et la meilleure approximation des fonctions analytiques d’une variable réelle, (1926), Gauthier-Villars Paris · JFM 52.0256.02
[2] Gelfand, I.M; Shilov, G.E, ()
[3] Hardy, G.H; Littlewood, J.E, Some properties of fractional integrals, I, Math. Z., 27, 565-606, (1927) · JFM 54.0275.05
[4] de Ferriet, J.Kampe, Heat equation and Hermite polynomials, () · Zbl 0117.29901
[5] Lorentz, G.G, Approximation of functions, (1966), Holt, Rinehart & Winston New York · Zbl 0153.38901
[6] Mhaskar, H.N, On weighted polynomial approximation on the whole real line and related topics, () · Zbl 0516.41004
[7] Mhaskar, H.N; Mhaskar, H.N, Weighted polynomial approximation of entire functions, II, J. approx. theory, J. approx. theory, 33, 59-68, (1981) · Zbl 0475.41003
[8] Nikolskii, S.M, Approximation of functions of several variables and imbedding theorems, (1975), Springer Berlin/New York
[9] Paley, R.E.A.C; Wiener, N, Fourier transforms in the complex domain, () · Zbl 0008.15203
[10] Varga, R.S, On an extension of a result of S. N. Bernstein, J. approx. theory, 1, 176-179, (1968) · Zbl 0177.08803
[11] Widder, D.V, The heat equation, (1975), Academic Press New York · Zbl 0322.35041
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