A FUNCTION IN THE DIRICHLET SPACE SUCH THAT ITS FOURIER-SERIES DIVERGES ALMOST EVERYWHERE

An analytic function F on the disc belongs to B if \\F\\B = integral-1/0 integral-2pi/0 \F'(re(itheta)\dthetadr < infinity. Notice that B subset-of/not-equal H-1 subset-of/not-equal, where H-1 is the Hardy space of all analytic functions F so that \\F\\H-1 = sup 0<r<1 integral-2pi/0 \F(re(itheta))\dtheta < infinity, L1 is the Lebesgue space of integrable functions on [theta, 2pi], and the inclusion H-1 subset-of/not-equal L1 is taken in the sense of boundary values, that is, F is-an-element-of H-1 double-line arrow pointing right lim(r-->1-) RF(re(itheta) is-an-element-of L1. Kolmogorov in 1923 showed that there exists an f in L1 so that its Fourier series diverges almost everywhere. In 1953 Sunouchi showed that there exists an f in H-1 with an almost everywhere divergent Fourier series. The purpose of this note is to announce.