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Recently, it has been shown that the problem of rational approximation to e~~ in [O, + <*>) arises naturally in numerical methods for approximating solutions of heat-conduction-type partial differential equations [l] , [2]. This special approximation problem leads one to the general question of approximating functions on the half line [0, + 00 ). In this paper we wish to announce two results of this study which are in the spirit of work done by S. N. Bernstein. A complete description of this work with proofs of these results and additional results will appear elsewhere. In order to state these results, we need the following notation. For any nonnegative integer w, let irm denote the collection of all real polynomials of degree at most m. For given r>0 and s> 1, let S(r, s) denote the unique open ellipse in the complex plane with foci at x = 0 and x = r and semimajor and semiminor axes a and b such that b/a = (s — l)/(s+l). Finally, if f(z) is any entire function, we set

transactions of the american mathematical society

Converse theorems and extensions in Chebyshev rational approximation to certain entire functions in $[\ast \ast \ast w(\ast \ast 0,\,+\infty )\ast \ast \ast w\ast \ast $