Transformation of Fourier series by means of general monotone sequences. (English. Russian original) Zbl 1442.42012
Math. Notes 107, No. 5, 740-758 (2020); translation from Mat. Zametki 107, No. 5, 674-692 (2020).
Summary: Estimates of the norms and the best approximations of the generalized Liouville-Weyl derivative in the two-dimensional case are obtained.
MSC:
| 42A16 | Fourier coefficients, Fourier series of functions with special properties, special Fourier series |
| 42A10 | Trigonometric approximation |
| 41A50 | Best approximation, Chebyshev systems |
References:
| [1] | Simonov, B. V.; Tikhonov, S. Yu, Embedding theorems in constructive approximation, Mat. Sb., 199, 9, 107-148 (2008) · Zbl 1172.46023 · doi:10.4213/sm3941 |
| [2] | Bernstein, S., On the best approximation of continuous functions by polynomials of a given degree, Comm. Soc. Math. Kharkow, 13, 2, 49-194 (1912) |
| [3] | DeVore, R.; Lorentz, G. G., Constructive Approximation (1993), Berlin: Springer-Verlag, Berlin · Zbl 0797.41016 |
| [4] | Konyushkov, A. A., The best approximations of the trigonometric polynomials and the Fourier coefficients, Mat. Sb., 44, 86, 53-84 (1958) · Zbl 0081.28402 |
| [5] | Timan, M. F., Inverse theorems of the constructive theory of functions in L_p spaces (1 < p < ∞), Mat. Sb., 46, 88, 125-132 (1958) · Zbl 0082.27803 |
| [6] | Marcinkiewicz, J., Sur quelques intégrales du type de Dini, Ann. Soc. Polon. Math., 17, 42-50 (1938) · Zbl 0020.01101 |
| [7] | Besov, O. V., On conditions for the derivatives of periodic functions to belong to L_p, Nauchn. Dokl. Vyssh. Shkoly Ser. Fiz.-Mat. Nauki, 1, 13-17 (1959) · Zbl 0095.27403 |
| [8] | Stechkin, S. B., On best approximation of conjugate functions by trigonometric polynomials, Izv. Akad. Nauk SSSR Ser. Mat., 20, 2, 197-206 (1956) · Zbl 0070.06501 |
| [9] | Timan, M. F., The imbedding of the \(L_p^{\left( k \right)}\) classes of functions, Izv. Vyssh. Uchebn. Zaved. Mat., 10, 61-74 (1974) · Zbl 0349.46027 |
| [10] | Simonov, B. V.; Tikhonov, S., On embeddings of the functional classes defined by constructive characteristics, Approximation and Probability, Banach Center Publ., 285-307 (2006), Warsaw: Polish Acad. Sci. Inst. Math., Warsaw · Zbl 1115.46029 |
| [11] | Szalay, I., On the best approximation of factorized Fourier series, Approximation Theory, 235-241 (1975), Dordrecht: Reidel, Dordrecht · Zbl 0313.42003 |
| [12] | Stepanets, A. I.; Zhukina, E. I., Inverse theorems for the approximation of (ψ, β)-differentiable functions, Ukr. Math. J., 41, 8, 953-959 (1989) · Zbl 0754.42002 · doi:10.1007/BF01058314 |
| [13] | Jumabayeva, A., Liouville-Weyl derivatives, best approximations, and moduli of smoothness, Acta Math. Hungar., 145, 2, 369-391 (2015) · Zbl 1363.42002 · doi:10.1007/s10474-015-0485-x |
| [14] | Tikhonov, S., Embedding results in questions of strong approximation by Fourier series, Acta Sci. Math. (Szeged), 72, 117-128 (2006) · Zbl 1121.42005 |
| [15] | Tikhonov, S., Trigonometric series with general monotone coefficients, J. Math. Anal. Appl., 326, 1, 721-735 (2007) · Zbl 1106.42003 · doi:10.1016/j.jmaa.2006.02.053 |
| [16] | Dyachenko, M.; Tikhonov, S., Convergence of trigonometric series with general monotone coefficients, C. R. Math. Acad. Sci. Paris, 345, 3, 123-126 (2007) · Zbl 1141.42004 · doi:10.1016/j.crma.2007.06.009 |
| [17] | Dyachenko, M.; Tikhonov, S., AHardy-Littlewood theorem for multiple series, J. Math. Anal. Appl., 339, 1, 503-510 (2008) · Zbl 1283.42031 · doi:10.1016/j.jmaa.2007.06.057 |
| [18] | Gorbachev, D.; Liflyand, E.; Tikhonov, S., Weighted Fourier inequalities: Boas’ conjecture in \({ℝ^n}\), J. Anal. Math., 114, 99-120 (2011) · Zbl 1246.42011 · doi:10.1007/s11854-011-0013-z |
| [19] | Tikhonov, S., On L_1-convergence of Fourier series, J. Math. Anal. Appl., 347, 2, 416-427 (2008) · Zbl 1257.42009 · doi:10.1016/j.jmaa.2008.05.048 |
| [20] | Dyachenko, M.; Tikhonov, S., Integrability and continuity of the functions represented by trigonometric series: coefficients criteria, Studia Math., 193, 3, 285-306 (2009) · Zbl 1169.42001 · doi:10.4064/sm193-3-5 |
| [21] | Tikhonov, S., On generalized Lipschitz classes and Fourier series, Z. Anal. Anwendungen, 23, 4, 745-764 (2004) · Zbl 1065.26005 · doi:10.4171/ZAA/1220 |
| [22] | Tikhonov, S. Yu, Generalized Lipschitz classes and Fourier coefficients, Mat. Zametki, 75, 6, 947-951 (2004) · Zbl 1061.42002 · doi:10.4213/mzm563 |
| [23] | Liflyand, E.; Tikhonov, S., A concept of general monotonicity and application, Math. Nachr., 284, 1083-1098 (2011) · Zbl 1223.26017 · doi:10.1002/mana.200810262 |
| [24] | Potapov, M. K.; Simonov, B. V.; Tikhonov, S. Yu, Fractional Moduli of Smoothness (2016), Moscow: Maks-Press, Moscow |
| [25] | Potapov, M. K., On the equivalence of convergence criteria for Fourier series, Mat. Sb., 68, 110, 111-127 (1965) · Zbl 0151.07003 |
| [26] | Nikol’skii, S. M., Approximation of Functions of Several Variables and Embedding Theorems (1977), Moscow: Nauka, Moscow · Zbl 0496.46020 |
| [27] | Zygmund, A., Trigonometric Series (1959), Cambridge: Cambridge Univ. Press, Cambridge · Zbl 0085.05601 |
| [28] | Potapov, M. K., Approximation by ‘angle’, Proceedings of the Conference on the Constructive Theory of Functions, Budapest, 1969, 371-399 (1972), Budapest: Akadémiai Kiadó, Budapest |
| [29] | Besov, O. V.; Il’in, V. P.; Nikol’skii, S. M., Integral Representations of Functions and Embedding Theorems (1975), Moscow: Nauka, Moscow · Zbl 0352.46023 |
| [30] | A. Jumabayeva and B. Simonov, “Liouville-Weyl derivatives of double trigonometric series,” in Appl. Numer. Harmonic Anal.. (in press) · Zbl 1413.42005 |
| [31] | Potapov, M. K.; Simonov, B. V.; Tikhonov, S. Yu, Relations between mixed moduli of smoothness, and embedding theorems for Nikol’skii classes, Trudy Mat. Inst. Steklova, Vol. 269: Function Theory and Differential Equations, 204-214 (2010), Moscow: MAIK Nauka/Interperiodica, Moscow · Zbl 1213.46032 |
| [32] | B. V. Simonov, On the Question of whether the Transformed Fourier Series Belongs to the Space L_p, Available from VINITI, No. 4985-81 (1981) [in Russian]. |
| [33] | M. G. Esmaganbetov, Conditions for the Existence of Mixed Weyl Derivatives in L_p ([0, 2π^2]) (1 < p< ∞) and Its Structural Properties, Available from VINITI, No. 1675-82 (1982) [in Russian]. |
| [34] | Potapov, M. K., The study of certain classes of functions by means of ‘angular’ approximation, Trudy Mat. Inst. Steklova, Vol. 117: Investigations in the Theory of Differentiable Functions of Many Variables and Its Applications. Part IV, 256-291 (1972), Moscow: MIAN, Moscow · Zbl 0293.41020 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
