Degree of approximation of functions in the Hölder metric (Q2757056)

Let \(A=(a_{i,j})_{i,j=0\cdots\infty}\) be an infinite matrix and let us define the A-Transform \(f\to t_{n}(f)=\sum_{k=0}^{\infty}a_{n,k}S_k(f)\), where \(f\in C_{2\pi}\) is an arbitrary continuous \(2\pi\)-periodic function and \(S_k(f)\) denotes the \(k-th\) partial sum of its Fourier series. The authors study the convergence behaviour of the differences \(t_{n}(f)-f\), in the norm of the space of Hölder functions \(H_{\beta}=\{f\in C_{2\pi} : |f(x)-f(y) |\leq K_f |x-y|^{\beta}\) for all \(x,y\in\mathbb{R} \}\), for functions \(f\in H_{\alpha}\) (\(\alpha>\beta\)) and different choices of the matrix \(A\).