О разных модулях гладкости и $K$-функционалах

Р. М. Тригуб

doi:10.37863/umzh.v72i7.2384

Р. М. Тригуб Донецк. нац. ун-т

DOI: https://doi.org/10.37863/umzh.v72i7.2384

Анотація

УДК 517.5

Статтю, що має оглядовий характер, присвячено визначенню точного порядка наближення функцiй лiнiйними середнiми рядiв i iнтегралiв Фур’є i знаходженню $K$-функцiоналiв через спецiальнi модулi гладкостi.

Посилання

A. F. Timan, Теория приближения функций действительного переменного (Russian) [[Teoriya priblizheniya funkczij dejstvitel`nogo peremennogo, Fizmatgiz, Moskva (1960)

O. V. Besov, Investigation of a family of functions spaces in connection with theorems of imbedding and extension, Tr. Mat. Inst. Steklovа, 60, 42 – 81 (1961).

R. Trigub, E. Belinsky, Fourier analysis and approximation of functions, Kluwer Academic Publishers, Dordrecht, xiv+585 pp. ISBN: 1-4020-2341-3 (2004), https://doi.org/10.1007/978-1-4020-2876-2 DOI: https://doi.org/10.1007/978-1-4020-2876-2

R. A. De Vore, G. G. Lorentz, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 303, Springer-Verlag, Berlin, x+449 pp. ISBN: 3-540-50627-6 (1993), https://doi.org/10.1007/978-3-662-02888-9 DOI: https://doi.org/10.1007/978-3-662-02888-9

V. K. Dzyadyk, I. A. Shevchuk, Theory of uniform approximation of functions by polinomials, Walter de Gruyter, Berlin; New York, xvi+480 pp. ISBN: 978-3-11-020147-5 (2008), https://doi.org/10.1515/9783110208245 DOI: https://doi.org/10.1515/9783110208245

M. F. Timan, О разностных свойствах функций многих переменных (Russian) [[O raznostny`kh svojstvakh funkczij mnogikh peremenny`kh]], Izv. AN SSSR, ser. mat, 33, №. 3, 667 – 676 (1969)

Yu. A. Brudnyi, Continuation of a function with preservation of order of decrease of moduli of continuity (Russian), Studies in the theory of functions of several variables, 33 – 53, Yaroslav. Gos. Univ., Yaroslavlʹ, (1980)

S. N. Bernshtejn, О свойствах однородных функциональных классов (Russian) [[O svojstvakh odnorodny`kh funkczional`ny`kh klassov]], Dokl. AN SSSR, 57, 111 – 114 (1947)

J. Marcinkiewics, A. Zygmund, On the differentiability of functions and summability of trigonometrical series, Fund. Math., 26, 1 – 43 (1936). DOI: https://doi.org/10.4064/fm-26-1-1-43

J. Boman, Equivalence of generalized modulus of continuity, Ark. Mat, 18, №. 1, 73 – 100 (1980), https://doi.org/10.1007/BF02384682 DOI: https://doi.org/10.1007/BF02384682

R. M. Trigub, Linear summation methods and the absolute convergence of Fourier series, Izv. Acad. Nauk SSSR, Ser. Mat., 32, 24 – 29 (1968).

I. Bergh, I. Löfström, ¨ Interpolation spaces, Grundlehren der Mathematischen Wissenschaften, No. 223. Springer-Verlag, Berlin-New York, x+207 pp. (1976). DOI: https://doi.org/10.1007/978-3-642-66451-9

Z. Ciesielski, Bases and $K$-functionals for Sobolev spaces on compact manifolds of class $C^{∞ }$ , Tr. Mat. Inst. Steklova, 164, 197 – 202 (1983).

R. M. Trigub, A formula for the $K$-functional of a couple of spaces of functions of several variables, Studies in the theory of functions of several real variables (Russian), 122 – 127, Matematika, Yaroslav. Gos. Univ., Yaroslavlʹ, (1988).

Z. Ditzian, A measure of smothness related to the Laplacian, Trans. Amer. Math. Soc., 326, 407 – 422 (1991), https://doi.org/10.2307/2001870 DOI: https://doi.org/10.2307/2001870

R. M. Trigub, Constructive characterizations of some functions classes, Izv. Akad. Nauk SSSR, Ser. Mat., 29, 615 – 630 (1965).

V. V. Zhuk, On approximation of periodic functions by linear methods of Fourier series, Dokl. Akad. Nauk SSSR, 173, 30 – 33 (1967).

V. V. Zhuk, Approximation of periodic functions, Leningrad. Univ. Press, Leningrad, 367 pp. (1982) (in Russian).

E. A. Storozhenko, On a problem of Hardy-Littlewood, Mat. Sb., 119, 564 – 583 (1982).

M. F. Timan, V. G. Ponomarenko, On approximation of periodic functions of two variables by summs of Marcinkiewicz type, Izv. Vyssh. Uchebn. Zaved. Mat., 9, 59 – 67 (1975).

E. S. Belinsky, Approximation by the Bochner-Riesz means and spherical modulus of continuity, Dop. Akad. Nauk Ukr. RSR, Ser. A, 7, 579 – 581 (1975).

R. M. Trigub, Absolute convergence of Fourier integrals, summability of Fourier series, and polynomial approximation of functions on the torus, Izv. Akad. Nauk SSSR, Ser. Math, 44, 1318 – 1409 (1980).

O. I. Kuznetsova, R. M. Trigub, Two-sided estimates of the approximation of functions by Riesz and Marcinkiewicz means, Dokl. Akad. Nauk SSSR, 251, № 1, 134 – 36 (1980).

Yu. L. Nosenko, Approximative properties of the Riesz means of double Fourier series, Ukr. Mat. Zh., 31, №. 1, 157 – 165 (1979).

Z. Ditzian, K. G. Ivanov, Strong converse inequalities, J. Anal. Math., 61, 61 – 111 (1993), https://doi.org/10.1007/BF02788839 DOI: https://doi.org/10.1007/BF02788839

B. R. Draganov, Exact estimates of the rate of approximation on convolution operators, J. Approxim. Theory, 162, No. 5952 – 979 (2010), https://doi.org/10.1016/j.jat.2009.10.003 DOI: https://doi.org/10.1016/j.jat.2009.10.003

R. M. Trigub, Multipliers in the Hardy spaces $H_p(D^m)$ for $p ∈ (0,1]$ and approximation properties of methodes for the summation of power series, Mat. Sb., 188, 145 – 160 (1997), https://doi.org/10.1070/SM1997v188n04ABEH000221 DOI: https://doi.org/10.1070/SM1997v188n04ABEH000221

A. V. Tovstolis, Fourier multipliers in Hardy spaces in tube domains over open cones and their applications, Methods Funct. Anal. and Topol., 4, 68 – 89 (1998).

Vit. V. Volchkov, Multipliers of power series in Hardy spaces, Ukr. Mat. Zh., 50, №. 5, 585 – 587 (1998), https://doi.org/10.1007/BF02487396 DOI: https://doi.org/10.1007/BF02487396

H. S. Shapiro, Some Tauberian theorem with applications to approximation theory, Bull. Amer. Math. Soc., 74, 500 – 504 (1968), https://doi.org/10.1090/S0002-9904-1968-11980-9 DOI: https://doi.org/10.1090/S0002-9904-1968-11980-9

E. M. Stein, Singular integrals and differentiability properties of functions, No. 30, Princeton Univ. Press, Princeton xiv+290 pp.(1970) DOI: https://doi.org/10.1515/9781400883882

E. M. Stein, G. Weiss, Introduction to Fourier analysis on Euclidean spaces, No. 32., Princeton Univ. Press, Princeton x+297 pp.(1971) DOI: https://doi.org/10.1515/9781400883899

B. М. Makarov, A. N. Podkorytov, Real analysis: measures, integrals and applications, Translated from the 2011 Russian original. Universitext. Springer, London, xx+772 pp. ISBN: 978-1-4471-5121-0; 978-1-4471-5122-7 (2013), https://doi.org/10.1007/978-1-4471-5122-7

E. Liflyand, S. Samko, R. Trigub, The Wiener algebra of absolutely convergent Fourier integrals: an overview, J. Anal. Math. Phys., No. 1, 1 – 68 (2012), https://doi.org/10.1007/s13324-012-0025-6 DOI: https://doi.org/10.1007/s13324-012-0025-6

R. M. Trigub, Multipliers of Fourier series, Ukr. Mat. Zh., 43, №. 12, 1686 – 1693 (1991)., https://doi.org/10.1007/BF01066697 DOI: https://doi.org/10.1007/BF01066697

R. M. Trigub, Multipliers of Fourier and $K$-functionals of spaces smoothness functions, translated from Ukr. Mat. Visn. 2 , no. 2, 236 – 280 (2005), 296 Ukr. Math. Bull. 2 , no. 2, 239 – 284 (2005)

R. Askey, Summability of Jacobi series, Trans. Amer. Math. Soc, 179, 71 – 81 (1973), https://doi.org/10.2307/1996491 DOI: https://doi.org/10.2307/1996491

E. Liflyand, R. Trigub, Conditions for the absolute convergence of Fourier integrals, J. Approxim. Theory, 163, No. 4, 438 – 459 (2011), https://doi.org/10.1016/j.jat.2010.11.001 DOI: https://doi.org/10.1016/j.jat.2010.11.001

S. B. Stechkin, On the order of the best approximations of continuous functions, Izv. Akad. Nauk SSSR, Ser. Mat., 15, №. 3, 219 – 242 (1951).

Yu. S. Kolomoitsev, R. M. Trigub, On one nonclassical method of approximation of periodic functions by trigonometric polynomials, Ukr. Math. Bull., 9, №. 3 (2012)., https://doi.org/10.1007/s10958-012-1111-x DOI: https://doi.org/10.1007/s10958-012-1111-x

R. M. Trigub, The exact order of approximation to periodic functions by Bernstein – Stechkin polinomials, Sb. Math., 204, №. 12, 1819 – 1838 (2013), https://doi.org/10.1070/sm2013v204n12abeh004362 DOI: https://doi.org/10.1070/SM2013v204n12ABEH004362

O. V. Kotova, R. M. Trigub, Exact order of approximation of periodic functions by one nonclassical method of summation of Fourier series, Ukr. Math. J., 64, №. 7, 1090 – 1108 (2012), https://doi.org/10.1007/s11253-012-0701-6 DOI: https://doi.org/10.1007/s11253-012-0701-6

P. M. Tamrazov, Smoothness and polynomial approximation, Naukova Dumka, Kiev, 271 pp. (1975) (in Russian).

Yu. S. Kolomoitsev, Description of a class of functions with the condition $ω_r (f; h)_p ≤ Mh^{r - 1+ 1/p}$ for $p ∈ (0, 1)$, Vestnik Dnepr. Univ, Ser. Mat., 8, 31 – 43 (2003).

R. Trigub, Fourier multipliers and comparison of linear operators, Oper. Theory: Adv. and Appl., 191, 499 – 513 (2009), https://doi.org/10.1007/978-3-7643-9921-4_31 DOI: https://doi.org/10.1007/978-3-7643-9921-4_31

Yu. Kolomoitsev, S. Tikhonov, Properties of moduli of smoothness in Lp(BbbRd), arXiv.org > математика > arXiv:1907.12788v3[math.CA] 25 Mar 2020 DOI: https://doi.org/10.1016/j.jat.2020.105423

Yu. Kolomoitsev, On moduli of smoothness and averaged differences of fractional order, Fract. Calc. and Appl. Anal., 20, №. 4, 988 – 1009 (2017), https://doi.org/10.1515/fca-2017-0051 DOI: https://doi.org/10.1515/fca-2017-0051

Z. Ditzian, V. Totik, Moduly of smoothness, Berlin, New York, x+227 pp. ISBN: 0-387-96536-X (1987), https://doi.org/10.1007/978-1-4612-4778-4 DOI: https://doi.org/10.1007/978-1-4612-4778-4

R. M. Trigub, On Fourier multipliers and absolutely convergent of Fourier integrals of radial functions, Ukr. Math. Zh., 62, №. 9, 1280 – 1293 (2010).

V. V. Lebedev, A. M. Olevskii, $L_p$ -мультипликаторы Фурье с ограниченными степенями (Russian) [[$L_p$ -mul`tiplikatory` Fur`e s ogranichenny`mi stepenyami]], Izv. RAN, ser. mat., vy`p. 70, №. 3, 129 – 166 (2006)

O. V. Kotova, R. M. Trigub, Approximative properties of the summation methods of Fourier integrals, J. Math. Sci., 211, №. 5, 668 – 683 (2015), https://doi.org/10.1007/s10958-015-2623-y DOI: https://doi.org/10.1007/s10958-015-2623-y

Yu. A. Brudny`j, Исследование свойств непериодических функций многих переменных методами теории приближений (Russian) [[Issledovanie svojstv neperiodicheskikh funkczij mnogikh peremenny`kh metodami teorii priblizhenij]], Uspekhi mat. nauk, vy`p. 20, №. 5, 270 – 272 (1965)

A. Brudnyi, Y. Brudnyi, Methods of geometric analysis in extension and trace problems , Monogr. Math., 1, 170 – 178, xx+414 pp. ISBN: 978-3-0348-0211-6 (2011).

A. M. Shveczova, Приближение частными суммами Фурье и наилучшее приближение некоторых классов функций (Russian) [[Priblizhenie chastny`mi summami Fur`e i nailuchshee priblizhenie nekotory`kh klassov funkczij]], Anal. Math., 27, 201 – 222 (2001).

R. M. Trigub, Асимптотика приближения непрерывных периодических функций линейными средними их рядов Фурье (Russian) [[Asimptotika priblizheniya neprery`vny`kh periodicheskikh funkczij linejny`mi srednimi ikh ryadov Fur`e]], Izv. RAN, ser. mat., vy`p. 84, #. 3, 185 – 202 (2020)

K. M. Davis and Y.-C. Chang, Lectures on Bochner – Riesz means, London Mathematical Society Lecture Note Series, 114, Cambridge University Press, Cambridge, x+150 pp. ISBN: 0-521-31277-9(1987), https://doi.org/10.1017/CBO9781107325654 DOI: https://doi.org/10.1017/CBO9781107325654

R. M. Trigub, Exact order of approximation of periodic functions by linear polynomials operators, East J. Approxim., 15, № 1, 25 – 50 (2009).

E. S. Belinsky, Strong summability of periodic functions and embilding theorems, Dokl. Akad. Nauk, 332, 133 – 134 (1993).

V. Totik, Approximation by Bernstein polinomials, Amer. J. Math., 116, 995 – 1018 (1994), https://doi.org/10.2307/2375007 DOI: https://doi.org/10.2307/2375007

Jia-ding Cao, H. Gonska, D. Kacsó, On the impossibility of certaine lower estimates for sequences of linear operators, Math. Balkanica (N.S.), 19, № 1-2, 39 – 58 (2005).

R. M. Trigub, Approximation of functions by polynomials with various constraints, Izv. Nats. Akad. Nauk (N.S.) Armen., 44, №. 4, 35 – 52 (2009), https://doi.org/10.3103/S1068362309040049 DOI: https://doi.org/10.3103/S1068362309040049