Geodesic
Double cosine-sine series and Nikol'skii classes in uniform metric
Problemy analiza, Tome 8 (2019) no. 3, pp. 187-203.

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In the this paper, we give neccessary and sufficient conditions for a function even with respect to the first argument but odd with respect to the second one to belong to the Nikol'skii classes defined by a mixed modulus of smoothness of a mixed derivative (both have arbitrary integer orders). These conditions involve the growth of partial sum of Fourier cosine-sine coefficients with power weights or the rate of decreasing to zero of these coefficients. A similar problem for generalized "small" Nikol'skii classes is also treated.
Keywords: double cosine-sine series, mixed modulus of smoothness, Nikol'skii classes. 
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S. S. Volosivets. Double cosine-sine series and Nikol'skii classes in uniform metric. Problemy analiza, Tome 8 (2019) no. 3, pp. 187-203. http://geodesic.mathdoc.fr/item/PA_2019_8_3_a16/

[1] Bary N. K., Stechkin S. B., “Best approximations and differential properties of two conjugate functions”, Trudy Moskov. Mat. Obshch., 5, 1956, 483–522 (in Russian) | MR

[2] Boas R. P., “Fourier series with positive coefficients”, J. Math. Anal. Appl., 17:3 (1967), 463–483 | DOI | MR | Zbl

[3] Butzer P. L., Dyckhoff H., Görlich E., Stens R. L., “Best trigonometric approximations, fractional order derivatives and Lipschitz classes”, Canad. J. Math., 29:4 (1977), 781–793 | DOI | MR | Zbl

[4] Chan L. Y., “On Fourier series with non-negative coefficients and two problems of R. P. Boas”, J. Math. Anal. Appl., 110:1 (1985), 116–129 | DOI | MR | Zbl

[5] Donskikh S. L., “Multiple Fourier series for functions from a Zygmund class”, Moscow Univ. Math. Bull., 65:1 (2010), 1–9 | DOI | MR | Zbl

[6] Dyachenko M. I., “Trigonometric series with generalized monotone coefficients”, Izv. Vyssh. Uchebn. Zaved. Mat., 1986, no. 7, 39–50 | MR | Zbl

[7] Dyachenko M., Tikhonov S., “Smoothness and asymptotic properties of functions with general monotone Fourier coefficients”, J. Fourier Anal. Appl., 24:4 (2018), 1072–1097 | DOI | MR | Zbl

[8] Fülöp V., “Double cosine series with nonnegative coefficients”, Acta Sci. Math.(Szeged), 70:1–2 (2004), 91–100 | MR | Zbl

[9] Fülöp V., “Double sine and cosine-sine series with nonnegative coefficients”, Acta Sci. Math.(Szeged), 70:1–2 (2004), 101–116 | MR | Zbl

[10] Fülöp V., Móricz F., “Absolutely convergent multiple Fourier series and multiplicative Zygmund classes of functions”, Analysis, 28:3 (2008), 345–354 | DOI | MR | Zbl

[11] Han D., Li G., Yu D., “Double sine series and higher order Lipschitz classes of functions”, Bull. Math. Anal. Appl., 5:1 (2013), 10–21 | MR | Zbl

[12] Liflyand E., Tikhonov S., Zeltser M., “Extending tests for convergence of number series”, J. Math. Anal. Appl., 377:1 (2011), 194–206 | DOI | MR | Zbl

[13] Lorentz G. G., “Fourier-Koeffizienten und Funktionenklassen”, Math. Zeitschr., 51:2 (1948), 135–149 | DOI | MR | Zbl

[14] Móricz F., “Absolutely convergent multiple Fourier series and multiplicative Lipschitz classes of functions”, Acta Math. Hung., 121:1–2 (2008), 1–19 | DOI | MR | Zbl

[15] Németh J., “Note on the Fourier series with nonnegative coefficients”, Acta Sci. Math. (Szeged), 55:1–2 (1991), 83–93 | MR | Zbl

[16] Németh J., “On the Fourier series with nonnegative coefficients”, Acta Sci. Math. (Szeged), 55:1–2 (1991), 95–101 | MR | Zbl

[17] Tr. Mat. Inst. Steklova, 2003, 244–256 | MR | Zbl

[18] Potapov M. K., Simonov B. V., Tikhonov S. Yu., “Mixed moduli of smoothness in $L_p$, $1

\infty$: a survey”, Surveys in Appr. Theory, 8 (2013), 1–57 | MR | Zbl

[19] Rubinstein A. I., “On $\omega$-lacunary series and functions from classes $H^\omega$”, Mat. sb., 65(107):2 (1964), 239–271 (in Russian) | MR | Zbl

[20] Tevzadze T. Sh., “On certain classes of functions and Fourier series”, Trudy Tbilisskogo universiteta (Proceedings of Tbilisi Univ.), 149 (1973), 150, 51–64 (in Russian) | MR

[21] Tikhonov S., “Smoothness conditions and Fourier series”, Math. Ineq. Appl., 10:2 (2007), 229–242 | DOI | MR | Zbl

[22] Tikhonov S., “On generalized Lipschitz classes and Fourier series”, Zeichr. Anal. Anwend., 23:4 (2004), 745–764 | DOI | MR | Zbl

[23] Tikhonov S., “Trigonometric series of Nikol'skii classes”, Acta Math. Hungar., 114:1–2 (2007), 61–78 | DOI | MR | Zbl

[24] Tikhonov S., “Best approximation and moduli of smoothness: computation and equivalence theorems”, J. Approx. Theory, 153:1 (2008), 19–39 | DOI | MR | Zbl

[25] Timan A. F., Theory of approximation of functions of a real variable, Macmillan, New York, 1963 | MR

[26] Volosivets S. S., “Multiple Fourier coefficients and generalized Lipschits classes in uniform metric”, J. Math. Anal. Appl., 427:2 (2015), 1070–1083 | DOI | MR | Zbl

[27] Volosivets S. S., “Multiple sine series and Nikol'skii classes in uniform metric”, European J. Math., 5:1 (2019), 206–222 | DOI | MR | Zbl

[28] Yu D., “Double trigonometric series with positive coefficients”, Anal. Math., 35:2 (2009), 149–167 | DOI | MR | Zbl

[29] Zygmund A., Trigonometric series, v. 2, Cambridge Univ. Press, Cambridge, 1959 | MR | Zbl