Móricz Ferenc honlapja

Személyi adatok | Publikációk | Oktatási tevékenységek


Publikációs lista

Vissza.

Cím:Lebesgue integrability and $L\sp 1$-convergence of trigonometric series with special coefficients.
Szerző:M\'oricz, F.
Forrás:Approximation theory, Proc. Conf., Kecskem\'et/Hung. 1990, Colloq. Math. Soc. J\'anos Bolyai 58, 513-536 (1991).
Nyelv:English
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) 
Letöltés:  | Zentralblatt