Convergence of Fourier Series - Uniform Convergence

Uniform Convergence

Suppose, and has modulus of continuity (we assume here that is also non decreasing), then the partial sum of the Fourier series converges to the function with the following speed

(It would be nice if the author can give the proof or cite it)

for a constant that does not depend upon, nor, nor .

This theorem, first proved by D Jackson, tells, for example, that if satisfies the -Hölder condition, then

If is periodic and absolutely continuous on, then the Fourier series of converges uniformly, but not necessarily absolutely, to, see p. 519 Exercise 6 (d) and p. 520 Exercise 7 (c), Introduction to classical real analysis, by Karl R. Stromberg, 1981.