On the power integrability with a weight of trigonometric series from $RBVS_{+,\omega}^{r,\delta }$ class


  • Xhevat Krasniqi University of Prishtina


$L^p-$integrability, Trigonometric series, Almost monotone sequence, Rest bounded variation sequence, Modulus of continuity


In this article, we have presented the necessary and sufficient conditions for the
power integrability with a weight of the sum of sine and cosine series whose
coefficients belong to the $RBVS_{+,\omega}^{r,\delta }$ class.


R.P. Boas, Jr., Integrability of trigonometrical series III, Quart. J. Math. (Oxford), 3 (1952), no. 2, 217-221.

M. Dyachenko and S. Tikhonov, Integrability and continuity of functions represented by trigonometric series: coefficients criteria, Studia Math., 193 (2009), no. 3, 285-306.

Heywood, On the integrability of functions defined by trigonometric series, Quart. J. Math. (Oxford), 5 (1954), no. 2, 71-76.

Xh. Z. Krasniqi, Rest bounded second variation sequences and $p$-th power integrability of some functions related to sums of formal trigonometric series, Acta Math. Acad. Paedagog. Nyhazi (N.S.), 31 (2015), no. 2, 249-257.

Xh. Z. Krasniqi and B. Szal, On the integrability with weight of trigonometric series, Izv. Nats. Akad. Nauk Armenii Mat., 55 (2020), no. 3, 57-67.

L. Leindler, A new class of numerical sequences and its applications to sine and cosine series, Analysis Math., 28 (2002), 279-286.

L. Leindler, Two embedding theorems, Acta Sci. Math. (Szeged), 71 (2005), no. 3-4, 619-630.

L. Leindler, Über verschiedene Konvergenzarten trigonometrischer Reihen. III. Bedingungen in der Metrik von $Lsp{p}$.}, (in German) Acta Sci. Math. (Szeged), 27 (1966), 205-215.

J. Németh, Power-monotone sequences and integrability of trigonometric series, JIPAM. J. Inequal. Pure Appl. Math., 4 (2003), no. 1, Article 3, 6 pp.

M. K. Potapov, A certain imbedding theorem, (in Russian) Mathematica (Cluj), 14 (1972), no. 37, 123-146.

B. Szal, On weighted $L^{p}$ integrability of functions defined by trigonometric series, J. Inequal. Appl., 2010 (2010), 1-19.

S. Yu. Tikhonov, On the integrability of trigonometric series, (in Russian) Mat. Zametki, 78 (2005), no. 3, 476-480; translation in Math. Notes, 78 (2005), no. 3-4, 437--442.

S. Yu. Tikhonov, Trigonometric series with general monotone coefficients, J. Math. Anal. Appl., 326 (2007), 721-735.

D. S. You, P. Zhou and S. P. Zhou, On $L^{p}$ integrability and convergence of trigonometric series, Studia Math., 182 (2007), no. 3, 215-226.

W. H. Young, On the Fourier series of bounded variation, Proc. London Math. Soc., 12 (1913), 41-70.




How to Cite

Krasniqi, X. (2020). On the power integrability with a weight of trigonometric series from $RBVS_{+,\omega}^{r,\delta }$ class. Armenian Journal of Mathematics, 12(8). Retrieved from http://armjmath.sci.am/index.php/ajm/article/view/409