On \(L^ 1\)-convergence of certain trigonometric sums. (English) Zbl 0696.42005

Let \(\sum^{\infty}_{k=1}a_ k\phi_ k(x)\), where \(\phi_ k(x)\) is cos kx or sin kx be a trigonometric series. Let the partial sum of this series be denoted by \(S_ n(x)\) and \(t(x)=\lim_{n\to \infty}S_ n(x).\) If \(a_ k=o(1)\), \(k\to \infty\) and \(\sum^{\infty}_{k=1}k^ 2| \Delta^ 2(\frac{a_ k}{k})| <\infty,\) we say that the series \(\sum^{\infty}_{k=1}a_ k\phi_ k(x)\) belongs to the class R. The authors introduce two new modified cosine and sine sums as \(f_ n(x)=\frac{a_ 0}{2}+\sum^{n}_{k=1}\sum^{n}_{j=k}\Delta (\frac{a_ j}{j})k\quad \cos kx\) and \(g_ n(x)=\sum^{n}_{k=1}\sum^{n}_{j=k}\Delta (\frac{a_ j}{j})k\quad \sin kx\) and study their \(L^ 1\)-convergence. Representing \(f_ n(x)\) or \(g_ n(x)\) by \(t_ n(x)\), the authors prove the following
Theorem: Let \(\sum^{\infty}_{k=1}a_ k\phi_ k(x)\) be in the class R. Then \(\lim_{n\to \infty}t_ n(x)=t(x)\) for \(x\in (0,\pi]\) and \(t\in L(0,\pi]\); moreover \(\| t_ n-t\| =o(1)\), \(n\to \infty\). They further deduce that if \(\sum a_ k\phi_ k(x)\) is in the class R, then \(\| t-S_ n\| =o(1)\), \(n\to \infty\) if and only if \(| a_{n+1}| \log n=o(1)\), \(n\to \infty\).
Reviewer: B.Ram


42A20 Convergence and absolute convergence of Fourier and trigonometric series
42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series