Approximation to conjugate functions

Abstract

The object of the present note is the establishment of the following theorem: THEOREM 1. If { Tn(x) } is a sequence of trigonometric polynomials of order n, and if (1) f(x) T.(x) = O(n-a) (a > 0) uniformly in x, then the conjugate function and the conjugate trigonometric polynomials satisfy (2) f(x) T(x) = O(n-a log n) uniformly in x. Furthermore the latter order is in general the best possible. For the sequence { Tn(x) } the sequence of partial sums of the Fourier series of f(x), Salem and Zygmund [3]1 showed that f(x) -s.(x) =0(n-a) for ax>0 uniformly in x implied that f(x) -s9(x) =O(n-a). Kawata [2] pointed out that for the sequence of Fejer means of the Fourier series of f(x) and for 0 1, of course, relation (3) implies that f(x) is a constant. Suppose first that 0 <a < 1. If condition (1) is satisfied, then by a theorem of S. Bernstein [1 ] f(x) ELip a. Hence, by another result of Bernstein [1], relation (3) holds and thus (4) follows. The polynomial Qn(x) = Tn(x) -on(x) has I Qn(x) I <Kn-a uniformly with respect to x. Hence, there is a constant C such that (6) | Qn(x) I Cn a log n. The combination of (4) and (6) gives (2). The argument for a( ? 1 proceeds in a similar fashion to that used Presented to the Society, June 17, 1950; received by the editors March 7,1950. 1 Numbers in brackets refer to the references at the end of the paper. 207