Rational Interpolation Function Research Articles

On approximation by bounded analytic functions

Introduction. Although this topic has been treated in a number of papers [2; 5; 6; 9; 13], important further progress is now possible thanks to a highly useful theorem on the conformal mapping of multiply connected regions [12]; this new theorem is used in combination with suitable refinements of older methods. Previous results have dealt primarily with rather rough (geometric) degrees of convergence and their relations to regions of convergence and regions of analyticity of functions approximated. The new theorem on conformal mapping combined with new results [13] on series of interpolation of rational functions enables us to treat here more refined degrees of convergence and their relations to regions of analyticity and to smoothness (e.g., Lipschitz conditions) on the boundary, of functions approximated. We consider (??1-6) degree of approximation on the common boundary of a region in which the approximating functions are analytic and of a smaller region in which the approximated function is analytic, a problem which is relatively new [13], and consider also (?7) measure of approximation on a suitable curve or set of curves interior to a given region in which the functions are analytic, an older problem [2]. A function f(z) analytic in a one-sided neighborhood of a Jordan curve C and continuous on C is said to be of class L(p, a) on C, where p ( ? 0) is integral and 0 <a < 1, if f(z) has a one-dimensional pth derivative on C which satisfies there a Lipschitz condition of order a. If C is analytic or even fulfills merely mild geometric conditions, then f(P) (z) exists [7, Theorem 2.2] also as a twodimensional derivative. Of course f(O) (z) =f(z). The definition just given has no meaning if p <0; for such integral values of p, 0 <a < 1, if C is an analytic Jordan curve, we say that f(z) is of class L(p, a) on C provided f(z) is analytic in a one-sided neighborhood of C and C can be expressed as the level locus u(z) = 1 of a nonconstant function u(z) harmonic in an annulus containing C and having no critical point in the annulus, and where in the neighborhood of C we have If(z) I < M(1 -p)p+a on the locus u(z) = p, po p < 1, po < 1; here M is to be independent of z and p. To be sure, this requirement is a restriction on the behavior of f(z) not on C but in a one-sided neighborhood of C. Nevertheless, as Hardy and Littlewood have shown, if C is the unit circle, and as also is true for C an arbitrary