Convergence of Fourier Series - Absolute Convergence

Absolute Convergence

A function ƒ has an absolutely converging Fourier series if

Obviously, if this condition holds then converges absolutely for every t and on the other hand, it is enough that converges absolutely for even one t, then this condition will hold. In other words, for absolute convergence there is no issue of where the sum converges absolutely — if it converges absolutely at one point then it does so everywhere.

The family of all functions with absolutely converging Fourier series is a Banach algebra (the operation of multiplication in the algebra is a simple multiplication of functions). It is called the Wiener algebra, after Norbert Wiener, who proved that if ƒ has absolutely converging Fourier series and is never zero, then 1/ƒ has absolutely converging Fourier series. The original proof of Wiener's theorem was difficult; a simplification using the theory of Banach algebras was given by Israel Gelfand. Finally, a short elementary proof was given by Donald J. Newman in 1975.

If belongs to a α-Hölder class for α > 1/2 then

for the constant in the Hölder condition, a constant only dependent on ; is the norm of the Krein algebra. Notice that the 1/2 here is essential—there are 1/2-Hölder functions which do not belong to the Wiener algebra. Besides, this theorem cannot improve the best known bound on the size of the Fourier coefficient of a α-Hölder function—that is only and then not summable.

If ƒ is of bounded variation and belongs to a α-Hölder class for some α > 0, it belongs to the Wiener algebra.