Fourier series: New results and unsolved problems
The present paper is a survey of results on some aspects of trigonometric approximation through summability methods together with an announcement of some new results and open problems. The results relate to functions in a specified class and the approximation is in the sense of the norm available in the class such as supremum norm, \(L^ p\) norm etc. A typical class is \(L^{\psi}_{\beta}\) defined as follows. Let \(\beta\) be a fixed real number and \(\psi\) (k) a sequence of real numbers such that \[ \sum^{\infty}_{k=1}(1/\psi (k))\{a_ k \cos (kx+\beta \pi /2)+b_ k \sin (kx+\beta \pi /2)\}, \] where \(a_ k\), \(b_ k\) are the Fourier coefficients of \(f\in L(0,2\pi)\), f being \(2\pi\)-periodic, is the Fourier series of a function in L(0,2\(\pi\)). Then f is said to belong to \(L^{\psi}_{\beta}\). The results and problems in the paper cannot be summarized here briefly since they require lot of symbolism.
trigonometric approximation through summability methods
scientific article; zbMATH DE number 4151069
scientific article