On everywhere divergence of trigonometric Fourier series

The following theorem is established. Theorem. Let a function phi : [0, +infinity) --> [0, +infinity) and a seque nce {phi(m)} satisfy the following condition: the function phi(u)/u is non- decreasing on (0, infinity), phi(m) greater than or equal to 1 (m = 1, 2,.. .), and phi(m)psi(m) = o(m root lnm/root lnlnm) as m --> infinity. Then the re is a function f is an element of L[-pi, pi] such that integral(-pi)(pi)phi(\ f(x)\) dx < infinity and lim sup(m-->infinity) S-m (f, x)/psi(m) =infinity for all x is an eleme nt of [-pi,pi]; here Sm(f) is the mth partial sum of the trigonometric Four ier series of f.