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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
##### Keywords:
heat equation; entire functions of exponential type
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##### References:
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