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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
Show Scanned Page ##### MSC:
 42A20 Convergence and absolute convergence of Fourier and trigonometric series
##### Keywords:
absolut convergent Fourier series