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