On approximations by trigonometric polynomials of classes of functions defined by moduli of smoothness. (English) Zbl 1470.42005

Berisha, Nimete Sh.; Berisha, Faton M.; Potapov, Mikhail K.; Dema, Marjan

Abstr. Appl. Anal. 2017, Article ID 9323181, 11 p. (2017).

Summary: In this paper, we give a characterization of Nikol’skiĭ-Besov type classes of functions, given by integral representations of moduli of smoothness, in terms of series over the moduli of smoothness. Also, necessary and sufficient conditions in terms of monotone or lacunary Fourier coefficients for a function to belong to such a class are given. In order to prove our results, we make use of certain recent reverse Copson-type and Leindler-type inequalities.

MSC:

42A10

Trigonometric approximation

References:

[1]	Laković, B., Ob odnom klasse funktsiĭ, Matematički Vesnik, 39, 4, 405-415, (1987) · Zbl 0681.42003
[2]	Tikhonov, S., Characteristics of Besov-Nikol’skiĭ class of function, Electronic Transactions on Numerical Analysis, 19, 94-104, (2005) · Zbl 1114.42002
[3]	Besov, O. V.; Il’in, V. P.; Nikol’skiĭ, S. M., Integral Representations of Functions and Imbedding Theorems. Integral Representations of Functions and Imbedding Theorems, Scripta Series in Mathematics, 2, (1979), New York, NY, USA: V. H. Winston & Sons, Washington, DC, USA; Halsted Press [John Wiley & Sons], New York, NY, USA · Zbl 0392.46023
[4]	Potapov, M. K.; Berisha, F. M.; Berisha, N. S.; Kadriu, R., Some reverse lp-type inequalities involving certain quasi monotone sequences, Mathematical Inequalities and Applications, 18, 4, 1245-1252, (2015) · Zbl 1355.26032 · doi:10.7153/mia-18-96
[5]	Tikhonov, S., Trigonometric series with general monotone coefficients, Journal of Mathematical Analysis and Applications, 326, 1, 721-735, (2007) · Zbl 1106.42003 · doi:10.1016/j.jmaa.2006.02.053
[6]	Liflyand, E.; Tikhonov, S., A concept of general monotonicity and applications, Mathematische Nachrichten, 284, 8-9, 1083-1098, (2011) · Zbl 1223.26017 · doi:10.1002/mana.200810262
[7]	Konyushkov, A. A., O klassakh lipshitsa, Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 21, 3, 423-448, (1957) · Zbl 0083.29101
[8]	Potapov, M. K.; Berisha, M. Q., Moduli of smoothness and the Fourier coefficients of periodic functions of one variable, Publications de l’Institut Mathématique (Beograd), 26, 40, 215-228, (1979) · Zbl 0429.42005
[9]	Hardy, G. H.; Littlewood, J. E.; Pólya, G., Inequalities. Inequalities, Cambridge Mathematical Library, (1988), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 0634.26008
[10]	Copson, E. T., Note on series of positive terms, Journal of the London Mathematical Society, 1-3, 1, 49-51, (1928) · JFM 54.0227.01 · doi:10.1112/jlms/s1-3.1.49
[11]	Leindler, L., Generalization of inequalities of Hardy and Littlewood, Acta Scientiarum Mathematicarum (Szeged), 31, 279-285, (1970) · Zbl 0203.06103
[12]	Leindler, L., Power-monotone sequences and Fourier series with positive coefficients, JIPAM Journal of Inequalities in Pure and Applied Mathematics, 1, 1, article 1, (2000) · Zbl 1006.42008
[13]	Tikhonov, S.; Zeltser, M., Weak monotonicity concept and its applications, Trends in Mathematics, 63, 357-374, (2014) · Zbl 1334.40004
[14]	Zygmund, A., Trigonometric Series, 2, (1988), Cambridge, UK: Cambridge University Press, Cambridge, UK · Zbl 0628.42001
[15]	Timan, A. F.; by, J., Theory of Approximation of Functions of a Real Variable. Theory of Approximation of Functions of a Real Variable, International Series of Monographs in Pure and Applied Mathematics, 34, (1963), New York, NY, USA: A Pergamon Press Book. The Macmillan Co., New York, NY, USA · Zbl 0117.29001
[16]	Berisha, M. Q.; Berisha, F. M., On monotone Fourier coefficients of a function belonging to Nikol’skiĭ-Besov classes, Mathematica Montisnigri, 10, 5-20, (1999) · Zbl 1013.42002