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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
MSC:
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
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