#### For citation:

Volosivets S. S., Vezhlev A. E. Embeddings of Generalized Bounded Variation Function Spaces into Spaces of Functions with Given Majorant of Average Modulus of Continuity. Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2017, vol. 17, iss. 3, pp. 255-266. DOI: 10.18500/1816-9791-2017-17-3-255-266

# Embeddings of Generalized Bounded Variation Function Spaces into Spaces of Functions with Given Majorant of Average Modulus of Continuity

In the present paper we study embeddings of some spaces of functions of generalized bounded variation into classes of functions with given majorant of average modulus of continuity introduced by B. Sendov and V. Popov. We consider the spaces ΛBV (p) of functions of bounded (Λ − p)-variation suggested by D. Waterman (for p = 1) and M. Shiba (for p > 1) and spaces V (v(n)) of functions with given majorant of its modulus of variation. The last quantity was introduced by Z. A. Chanturia. The necessary and sufficient conditions of such embeddings are proved. Earlier similar embeddings into classes with given majorant of usual integral modulus of continuity were studied by Yu. E. Kuprikov, U. Goginava and V. Tskhadaia, M. Hormozi et al. Applications of obtained results to estimates of errors for some quadrature rules are given.

- Jordan C. Sur la Serie de Fourier. C.R. Acad. Sci. Paris, 1981, vol. 92, pp. 228–230.
- Wiener N. The quadratic variation of a function and its Fourier coefficients. J. Math. and Phys., 1924, vol. 3, pp. 72–94.
- Hardy G. H., Littlewood J. E. Some properties of fractional integrals (I). Math. Zeitschr., 1928, vol. 28, pp. 565–606.
- Young L. C. An inequality of the Hölder type, connected with Stieltjes integration. Acta Math., 1936, vol. 67, pp. 251–282.
- Terekhin A. P. Priblizhenie funkcii ogranichennoi p-variacii [Approximation of functions of bounded p-variation]. Izvestiya vyssh. ucheb. zaved. Matematika, 1965, no. 2, pp. 171–187 (in Russian).
- Terekhin A. P. Integral smoothness properties of periodic functions of bounded p-variation. Math. Notes, 1967, vol. 2, no. 3, pp. 659–665.
- Waterman D. On convergence of Fourier series of functions of generalized bounded variation. Studia Math., 1972, vol. 44, no. 2, pp. 107–117.
- Waterman D. On the summability of Fourier series of functions of Λ-bounded variation. Studia math., 1976, vol. 55, no. 1, pp. 87–95.
- Shiba M. On absolute convergence of Fourier series of functions of class ΛBV (p) . Sci. Rep. Fukushima Univ., 1980, vol. 30, pp. 7–10.
- Chaturia Z. A. Absolute convergence of Fourier series. Math. Notes, 1975, vol. 18, no. 2, pp. 695–703.
- Kuprikov Yu. E. O modulyah nepreryvnosti funkcii iz klassov Watermana [On moduli of continuity of functions from Waterman classes]. Vestnik Mosk. univ. Ser. 1. Math., mech., 1997, no. 5, pp. 59–62 (in Russian).
- Li Z., Wang H. Estimates of L p - continuity modulus of ΛBV series and applications in Fourier series. Applicable Anal., 2011, vol. 90, no. 3–4, pp. 475–482.
- Hormozi M. Inclusion of ΛBV (p) spaces in the classes H q ω . J. Math. Anal. Appl., 2013, vol. 404, no. 1, pp. 195–200.
- Goginava U., Tskhadaia V. On the embedding V [v(n)] ⊂ H ω p . Proc. A. Razmadze Math. Inst., 2004, vol. 136, pp. 47–54.
- Sendov B., Popov V. Usrednennye moduli gladkosti [Average moduli of smoothness]. Мoscow, Mir, 1988. 328 p (in Russian).
- Hardy G. H., Littlewood J. E., Polya G. Inequalities. Cambridge, Cambridge Univ. Press, 1934. 328 p.

- 243 reads