zbMATH — the first resource for mathematics

On certain classes of absolut convergent Fourier series. (Sur certaines classes de séries de Fourier absolument convergents.) (French) Zbl 0072.06004
Let \(A\) denote the class of functions \(f(x)=\sum_{-\infty}^{\infty}a_{n}e^{inx}\) with \(||f||= \sum|a_{n}|<\infty\), and \(A\{\omega_{n}\}\) denote the class of functions \(F(x)= \sum_{-\infty}^{\infty}A_{n}e^{inx}\) with \(|||F|||=\sum |A_{n}|\omega_{n} <\infty\), where \(\{\omega_{n}\}_{-\infty}^{\infty}\) is a given sequence such that \(\omega_{n}\geq 1\) for all \(n\). \(A\) and \(A\{\omega_{n}\}\) are Banach spaces with the norms \(||f||\) and \(|||F|||\) respectively. \(A^{*}\) denotes the class of functions \(f\) which are locally equal to functions of \(A\). The author begins by proving the following extension of a theorem of Leibenson: If \(F\in A(\omega_{n})\), then \(F(f(x))\in A\) if and only If \(f\) is real and \(||e^{inf}||=O(\omega_{n})\), as \( n\rightarrow\pm\infty\). With the help of this result and a classical result of Wiener relating to the absolute convergence of Fourier series he deduces: If in the neighbourhood of every point \(x\), \( f\) is equal to a real function \(f_{x}\) for which \(||e^{inf_x}=O(\omega_{n})\) as \( n\rightarrow\pm\infty\), then \(||e^{\inf}||=O(\omega_{n})\), He uses these two results to prove a number of theorems concerning the rapidity of the growth of \(||e^{inf}||\) when \(f\) is either linear by intervals or analytic. The following two results are typical: (1) If \(f\) is real continuous and linear by intervals satisfying \(f(x+2\pi)=f(x) \pmod 2\pi\), then \(\| e^{inf}\|=O(\log n)\), \(n\to\pm\infty\); moreover if \(f(x)=|x|\) in \([-\pi,\pi]\), then \(\| e^{inf}\|=(2/\pi)\log n +O(1)\).(2) If \(f\) is a real, non-constant, analytic \(2\pi\)-periodic function, then there exist two positive constants \(\lambda_{1}\) and \(\lambda_{2}\) such that \(\lambda_{1}\sqrt{|n|}<\| e^{\inf}\|<\lambda_{2}\sqrt{|n|}\) for all integral \(n\). Several interesting corollaries are mentioned and finally an example \(f\) is given for which \(f\in A^{*}\) but \(|f|\not\in A^{*}\).
Reviewer: U. N. Singh

42A20 Convergence and absolute convergence of Fourier and trigonometric series