S. Bitimkhan. Conditions of absolute Cesaro summability of multiple trigonometric Fourier series ... P. 42-47

A necessary and sufficient condition of absolute $|C;\overline{\beta}|_\lambda$-summability almost everywhere on ${\mathbb T}^s$ is obtained for multiple trigonometric Fourier series of functions $f\in L_{\overline{q}}({\mathbb T}^s)$ from generalized Besov classes $B_{\overline q,s,\theta}^{\omega_r}$, where ${\mathbb T}^s=[0,2\pi)^s$, $\overline{\beta}=(\beta_1,\beta_2,\ldots,\beta_s)$, $\overline{q}=(q_1,q_2,\ldots, q_s)$, $1<q_j\le 2$, $\overline{1,s}$, $1\le \lambda\le q_s\le \ldots\le q_1$, $\lambda<\theta<\infty$, $0\le \beta_j<1/q'_j=1-1/q_j$, $\overline{1,s}$, $r\in \mathbb{N}$, $r>\sum_{j=1}^s(1/q_j-\beta_j)$, and $\omega_r$ is a function of the type of modulus of smoothness of order $r$.

Keywords: multiple trigonometric Fourier series, absolute summability, modulus of smoothness, generalized Besov class

Received August 31, 2018

Revised March 27, 2019

Accepted April 29, 2019

Samat Bitimkhan, Cand. Phys.-Math. Sci., Karagandy State University named after E.A.Buketov, Karagandy city, 100028 Kazakhstan, e-mail: bsamat10@mail.ru

REFERENCES

1.   Timan A.F. Theory of approximation of functions of real variables. Macmillan, Pergamon Press, 1963. 631 p. Original Russian text published in Timan A.F. Teorija priblizhenija funkcij dejstvitel’nogo peremennogo. Moscow: Fizmatgiz Publ., 1960, 624 p.

2.   Timan M.F. Absolute convergence and summability of Fourier series. Soobshch. Akad. Nauk Gruzin. SSR, 1961, vol. 26, no. 6, pp. 641–646 (in Russian).

3.   Ponomarenko Yu.A. Criteria for absolute Cesaro summability of multiple Fourier series. Dokl. Akad. Nauk SSSR, 1963, vol. 152, no. 6, pp. 1305–1307 (in Russian) .

4.   Ponomarenko Yu.A., Timan M.F. On the absolute summability of Fourier multiple series. Ukr. Math. J., 1971, vol. 23, no. 3, pp. 291–304. doi: 10.1007/BF01085351 

5.   Szalay I. Absolute summability of trigonometric series. Math. Notes, 1981, vol. 30, no. 6, pp. 912–919. doi: 10.1007/BF01145770 

6.   Timan M.F., Ponomarenko Yu.A. Some criteria of absolute summability of Fourier series. In: Issled. Sovrem. Probl. Konstr. Teor. Funkts., sbornik trudov, Baku: AN Azerb. SSR, 1965, pp. 489–492 (in Russian).

7.   Gol’dman M.L. On the inclusion of generalized HЈolder classes. Math. Notes, 1972, vol. 12, no. 3, pp. 626–631. doi: 10.1007/BF01093999 

8.   Akishev G.A., Bitimkhanuly S. Moduli of smoothness and the absolute summability of multiple trigonometric series. Mat. Zh. Almaty, 2003, vol. 3, no. 1, pp. 5–14 (in Russian).

9.   Bitimkhan S., Akishev G. The conditions of absolute summability of multiple trigonometric series. AIP Conf. Proc., 2015, vol. 1676, no. 1, 020095. doi: 10.1063/1.4930521 

10.   Ul’yanov P.L. Absolute and uniform convergence of Fourier series. Math. USSR-Sb., 1967, vol. 1, no. 2, pp. 169–197. doi: 10.1070/SM1967v001n02ABEH001973 

11.   Bitimkhanuly S. Condition for the absolute summability of multiple trigonometric series. Vestnik Kazakh. Univ. Ser. Mat., Mekh., Inf., 2001, no. 1 (24), pp. 3–11 (in Russian).

Cite this article as: S.Bitimkhan, Conditions of absolute Cesaro summability of multiple trigonometric Fourier series, Trudy Instituta Matematiki i Mekhaniki URO RAN, 2019, vol. 25, no. 2, pp. 42–47.