Optimal convergence factors for general Fourier coefficients

Abstract

Abstract Stefan Banach proved that even the Fourier series of the function f ⁢ ( x ) = 1 {f(x)=1} ( x ∈ [ 0 , 1 ] ) {(x\in[0,1])} might not be convergent for some orthonormal systems (ONS). Thus we can conclude that the Fourier series of functions belonging to a certain differential class cannot be convergent in general. On the other hand, for classical ONS (trigonometric, Haar, Walsh systems, …), the convergence of Fourier series of differentiable class functions is a simple subject. In this paper, we investigate the sequence of positive numbers such that, multiplying the Fourier coefficients of Lipschitz class functions by these numbers, one obtains a convergent series of the special form. From the convergence of these special form series, we derive the convergence of the special Fourier series of Lipschitz class functions with respect to general ONS. The obtained results are best possible.