On the category of the set of uniform convergence for both Fourier series and Lagrange interpolation process (Q1901951)

Let \(C_{2\pi}\) be the normed vector space of real, continuous and \(2\pi\)-periodic functions, endowed with uniform norm. \({\mathcal L}_n(f; \cdot)\) stands for Lagrange polynomial which interpolates a function \(f\) at equidistant points \(x_{k,n}:= 2\pi k(2n+ 1)^{- 1}\), where \(n\in \mathbb{N}\) and \(k\in -n,\dots, n\). For the function \(f\), \(S_nf\) is the \(n\)th partial sum of the Fourier series. Let \(U\) denote the space of all functions from \(C_{2\pi}\) which have uniformly convergent Fourier series, provided with the norm \(|f|_U:= \sup_n|S_n f|_C\). Let \(S\) be the normed vector space of functions \(f\in C_{2\pi}\) so that the interpolation process \(({\mathcal L}_n f)\) uniformly converges. The space \(S\) is endowed with the following norm \(|f|_S:= \sup_n |{\mathcal L}_n f|_C\). If \(A\) is a non-empty set of functions \(f\in C_{2\pi}\) such that \[ \lim_{n\to \infty} |{\mathcal L}_n f- f|_C= \lim_{n\to \infty} |S_n f- f|_C= 0, \] then the following is true: Theorem. The set \(A\) is a set of first Baire category both in the space \(U\) and in the space \(S\).