Generalized continuity of higher order symmetric derivatives
Let f be a real-valued function defined in a neighbourhood of x, let m be a positive integer. If there are real numbers \(\beta_{2k}\), \(0\leq k\leq m\), depending on x but not on h such that \[ (f(x+h)+f(x- h))=\sum^{m}_{i=0}\frac{h^{2i}}{(2i)!}\beta_{2i}+o(h^{2m}),\quad h\to 0, \] then \(\beta_{2m}\) is called the symmetric de la Vallée Poussin derivative of f at x of order 2m. Similarly if there are numbers \(\beta_{2i+1}\), \(0\leq i\leq m\), such that \[ (f(x+h)-f(x- h))=\sum^{m}_{i=0}\frac{h^{2i+1}}{(2i+1)!}\beta_{2i+1}+o(h^{2m+1 }),\quad h\to 0, \] then \(\beta_{2m+1}\) is called the symmetric de la Vallée Poussin derivative of f at x of order \(2m+1\). In the paper some very detailed continuity properties of such derivatives are given.
symmetric de la Vallée Poussin derivative
scientific article; zbMATH DE number 4154772