Approximation by integral functions in the complex domain

Abstract

The approximation by integral functions to functions defined on the real axis-oo <£< oo or on its positive part only (0<x< oo) has been treated by the author in an earlier paper(1). The present paper deals with the corresponding problem for functions defined on regions R of the extended z-plane (z=x+iy). The only case of interest is that in which the point 2= oo lies on the boundary of 7?, as has been shown in the previous paper. Originally the author had intended to treat only the cases where 7? is a half-plane or a strip. It has turned out, however, that simple results can be obtained for approximation in an angle 0 (0 < arg z < 6 ; 6 < 2w). This problem is evidently much more general than that concerning the half-plane, and it is adapted to elucidate results of the above paper. It will be shown that a function F(z), analytic and bounded in the interior of an angle 0 and satisfying some condition of continuity in the closed domain, can be approximated uniformly by integral functions of finite types of the order p=ir(2ir-0)~\ and that this is the "best possible" value (§2). The uniform approximation by integral functions to unbounded functions in an angle 0, in particular to zr (0<r< 00), is treated in §4; there are, however, rather interesting questions which arise in the general case, but are left open in the present investigation. The approximation by rational functions is known to apply under very restrictive conditions only; it fails, for instance, for the function zr. The approximation to bounded functions in the half-plane is treated in §3, that in the strip in §5. For the half-plane, extensive use is made of the fundamental theory of the class £>P(2), developed by E. Hille and J. D. Tamarkin; some of the results deduced for the strip can be regarded as an extension of this theory. The investigation of the strip starts from Theorem 13. Its proof and that of some other results is given as an appendix. Incidentally results

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