Absolute convergence of series of Fourier coefficients with respect to an orthonormal system
The paper under review is connected with the absolute convergence of the Fourier coefficients (or that of a subsequence) in terms of the modulus of continuity. S. B. Stechkin proved that for any subsequence \(\{m_ k\}\) of the natural numbers one has \[ \sum_ k(| a_{m_ k} |+| b_{m_ k} |) \leq C\sum{1\over \sqrt k} \omega^{(2)} \left({1 \over m_ k};f \right), \] where \(\omega^{(2)}\) is the \(L^ 2\)-modulus of continuity of \(f\). The author found in an earlier paper that the convergence of the left hand side for every function from the smoothness class \(H^ \omega\) is equivalent to the convergence of \(\sum{1\over \sqrt k} \omega \left({1\over m_ k} \right)\). In the present paper he extends these investigations to the absolute convergence of Fourier series with respect to arbitrary orthonormal systems on [0,1]. In particular, he obtains the complete analogue of the aforementioned result for general orthonormal systems under a mild and natural assumption on the modulus of continuity \(\omega\).
scientific article; zbMATH DE number 130688