Absztrakt: | [For the entire collection see Zbl 0746.00075.]\par
The author
considers the cosine series ${1\over 2}a\sb 0+\sum\sp \infty\sb{k=1}a\sb
k\cos kx \sim f(x)$, the sine series $\sum\sp \infty\sb{k=1}a\sb
k\sin kx$ $\sim g(x)$, and the complex trigonometric series $(*)\
\sum\sp \infty\sb{k=-\infty}c\sb ke\sp{ikx}\sim h(x)$ as well.
He presents very wide discussion of well-known results [see,
e.g., {\it A. Zygmund}, Trigonometric series. Vol. 1 (1959; Zbl
0085.056), p. 185; {\it G. A. Fomin}, Mat. Zametki 23, 213-222
(1978; Zbl 0379.42004); the author, Stud. Math. 92, No. 2, 187-200
(1989; Zbl 0671.42006); Proc. Am. Math. Soc. 113, No. 1, 53-64
(1991; Zbl 0728.42003)], giving new proofs. The following new
result is also proved:\par
Theorem 4. If $\lim\sb{\vert k\vert\to\infty}c\sb
k=0$ and $$\sum\sp \infty\sb{m=1}2\sp{m/q}\left(\sum\sb{\vert
k\vert\in[2\sp{m-1},2\sp m]}\vert c\sb k-c\sb{k\pm 1}\vert\sp
p\right)\sp{1/p}<\infty\quad\text{ for some } p>1,$$ then\par
(i)
$h\in L\sp 1\sb{(-\pi,\pi)}$ if and only if $\sum\sp \infty\sb{k=1}{\vert
c\sb k-c\sb{-k}\vert\over k}<\infty$,\par
(ii) if $h\in L\sp
1\sb{(-\pi,\pi)}$, then $(*)$ is the Fourier series of $h$,\par
(iii)
$\lim\sb{n\to\infty}\int\sp \pi\sb{-\pi}\left\vert\sum\sp n\sb{k=-
n}c\sb ke\sp{ikx}-h(x)\right\vert dx=0$ if and only if $\lim\sb{\vert
n\vert\to\infty}c\sb n\ln\vert n\vert=0$.}
\RV{R.Gajewski (Pozna\'n) |