On the summability of the differentiated Fourier series II

Abstract

In the previous paper [I] we established the following generalization of Fatou's theorem on the Abel summability of the differentiated Fourier series. Let f L(O, 2X) of period 2w with Afo(t) ==f(xo+t) -f(xo t) essentially bounded2 in a neighborhood of t = 0. Then if, for a = 2, a -fap,(xo) = y, the differentiated Fourier series of f is Abel summable to y at xo. We note that y =a-fap5(xo), i.e. y is the a-approximate symmetric derivative of f at x0, means that if, for any e > 0, H. = {t: |y-itx0(t)/2tI >e}, then m(H,C' (-t, t)) =o(ta) as t-)O. We showed there that a = 2 cannot be replaced by a smaller value and that essentially bounded cannot be omitted. Here we consider the question of replacing essentially bounded by a weaker condition. Clearly the differentiability condition plus the essential boundedness implies the condition