The convergence of Fourier series and a K-functional
For \(f\in B[-\pi,\pi]\) and I a subinterval of [-\(\pi\),\(\pi\) ] let \(\omega (f,I)=\sup_{t,u\in I}| f(t)-f(u)|.\) If [a,b]\(\subset [- \pi,\pi]\) and \(\delta >0\) let \(\Omega (f,\delta;[a,b])=\sup \delta \sum_{I\in {\mathcal I}}\omega (f,I),\) where the sup is taken over all families \({\mathcal I}\) of subintervals of [a,b] such that card(\({\mathcal I})\) \(=\) number of intervals in \({\mathcal I}\leq \delta^{-1}\), \(\cup_{I\in {\mathcal I}}I=[a,b]\) and the members of \({\mathcal I}\) are pairwise disjoint. The main result of the author is the following interesting theorem. Theorem. Let f be a bounded, measurable, \(2\pi\)-periodic function, \(x\in [-\pi,\pi]\) and \(f(x+)\), f(x-) exist. Then, for \(n\geq 1\), \[ | S_ n(f,x)-(1/2)(f(x+)+f(x-))| \leq C\sum^{n}_{k=1}(1/k)\Omega (g,k/n;[0,\pi /k]), \] where \(g(t)=f(x+t)+f(x-t)-f(x+)-f(x-)\) and \(S_ n(f)\) denotes the nth partial sum of the Fourier series of f(x).
scientific article; zbMATH DE number 4208764
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