Fourier coefficients and generalized Lipschitz classes in uniform metric
Absolutely convergent Fourier series \[ \sum_{k\in\mathbb{Z}} c_k e^{ikx}:= f(x),\quad\text{where }\sum_{k\in\mathbb{Z}} |c_k|< \infty, \] are considered, and equivalence relations are proved between the behavior of the Fourier coefficients \(c_k\) of a special kind and the smoothness properties of the sum function \(f\). The following Theorem 3 is typical: If \[ \sum_{|k|> n}|c_k|= o(n^{-m})\quad\text{for some }m\in\mathbb{N}, \] then the Schwarz derivative of \(f\) of order \(m\) exists at a given point \(x\) and equals \(A\) if and only if the formally differentiated series \[ \sum(ik)^m c_k e^{ikx}\tag{\(*\)} \] is convergent and its sum equals \(A\). Reviewer's remark: The reviewer proved in a joint paper with \textit{G. Brown} and \textit{Z. Sáfár} [Acta Sci. Math. (Szeged), 75, 161--173 (2009)] that an analogous characterization is valid for the existence of the ordinary derivative \(f^{((m)}(x)\) in place of the Schwarz derivative; and the following supplement is also valid: the derivative \(f^{(m)}\) is continuous on \(\mathbb{T}:= [-\pi,\pi)\) if and only if the formally differentiated series \((*)\) converges uniformly on \(\mathbb{T}\).
scientific article; zbMATH DE number 5578227