Integrated Continuity Conditions and Degree of Approximation by Polynomials or by Bounded Analytic Functions^)

Introduction. In the study of uniform approximation to a function of a complex variable by polynomials or by bounded analytic functions, Lipschitz conditions have proved extremely useful in relating degree of approximation to continuity properties of the functions approximated. Parts of this theory are analogous to the older study (S. Bernstein, D. Jackson, de la Vallee Poussin, Montel) of approximation to real periodic functions by trigonometric sums, where Lipschitz conditions have also proved useful. Hardy and Littlewood [2, p. 633 ](2) first pointed out that trigonometric approximation in the mean is likewise closely related to integrated Lipschitz conditions; proofs of the theorems stated by Hardy and Littlewood were first published by Quade [l]. The object of the present note is to indicate rapidly and with a minimum of detail, that degree of approximation in the mean by polynomials in the complex variable and by bounded analytic functions is also conveniently investigated by use of integrated Lipschitz conditions. The investigation leads naturally to the use also of classes of analytic functions satisfying integrated Zygmund and integral asymptotic conditions. We shall approximate functions on an analytic Jordan curve or set of curves in the z-plane. Throughout the paper 7 denotes the unit circle, C a single analytic Jordan curve or a finite number X of mutually exterior analytic Jordan curves Cy, j=l, 2, • • , X; if X = l, Ci = C. A function z = Xi(w), w = reie, maps the closed interior of y: \w\ =1 onto the closed interior of Cy one-to-one and conformally; w = fl,(z) denotes the function inverse to z = Xy(-0; Cy denotes the interior of Cy together with Cy. Similarly, w = cp(z) maps the exterior of C conformally (not necessarily one-to-one) onto | w\ > 1 so that 00 =1) is the image in the exterior of C ol \w\ =R under this mapping. In this paper, the letter s will denote arc-length measured on C, the letter p a number not less than unity, the letter k a non-negative integer unless otherwise indicated, the letters M and L with or without subscript constants independent of re, z and w. A function f(z) belongs to the class Hp on y if it is analytic interior to y and if Mp[f(rew)}={fln\f(reiB)Yd6}1'r> is bounded for 0<r<l. A function f(z)