## 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

### MSC:

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

### Keywords:

modified cosine sums; $$L^ 1$$-convergence