On L 1-Convergence of Fourier Series under the MVBV Condition

Yu, Dan Sheng ; Zhou, Ping ; Zhou, Song Ping

Canadian mathematical bulletin, Tome 52 (2009) no. 4, pp. 627-636

Résumé

Let $f\in \,{{L}_{2\pi }}$ be a real-valued even function with its Fourier series $\frac{{{a}_{0}}}{2}\,+\,\sum _{n=1}^{\infty }\,{{a}_{n}}\,\cos \,nx$ , and let ${{S}_{n}}\left( f,x \right)$ , $n\,\,\ge \,\,1$ , be the $n$ -th partial sum of the Fourier series. It is well known that if the nonnegative sequence $\{{{a}_{n}}\}$ is decreasing and ${{\lim }_{n\to \infty }}\,{{a}_{n}}\,=\,0$ , then $$\underset{n\to \infty }{\mathop{\lim }}\,{{\left\| f-{{S}_{n}}\left( f \right) \right\|}_{L}}=0\text{ifanyonlyif}\underset{n\to \infty }{\mathop{\lim }}\,{{a}_{n}}\log n=0.$$ We weaken the monotone condition in this classical result to the so-called mean value bounded variation (MVBV) condition. The generalization of the above classical result in real-valued function space is presented as a special case of the main result in this paper, which gives the ${{L}^{1}}$ -convergence of a function $f\in {{L}_{2\pi }}$ in complex space. We also give results on ${{L}^{1}}$ -approximation of a function $f\in {{L}_{2\pi }}$ under the MVBV condition.

Détail

DOI : 10.4153/CMB-2009-061-6

Mots-clés : 42A25, 41A50, complex trigonometric series, L 1 convergence, monotonicity, mean value bounded variation