Multiplicator type operators and approximation of periodic functions of one variable by trigonometric polynomials

K. V. Runovskii

Sbornik. Mathematics, Tome 212 (2021) no. 2, pp. 234-264

Résumé

The norms of the images of multiplier type operators generated by an arbitrary generator are estimated in terms of the best approximations of univariate periodic functions by trigonometric polynomials in the $L_p$-spaces, $1\le p\le+\infty$. As corollaries, estimates for the quality of approximation by Fourier means, an inverse theorem of approximation theory, comparison theorems, an analogue of the Marchaud inequality for generalized moduli of smoothness defined by a periodic generator, as well as some constructive sufficient conditions for generalized smoothness and Bernstein type inequalities for generalized derivatives of trigonometric polynomials are obtained. Bibliography: 49 titles.

Détail

Keywords: Fourier means, modulus of smoothness, generalized derivative, best approximation.
Mots-clés : multiplier

Bibliographie

[1] E. C. Titchmarsh, Introduction to the theory of Fourier integrals, Oxford, Clarendon Press, 1937, x+390 pp. | MR | Zbl

[2] E. M. Stein, G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Math. Ser., 32, Princeton Univ. Press, Princeton, NJ, 1971, x+297 pp. | MR | Zbl | Zbl

[3] E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Math. Ser., 30, Princeton Univ. Press, Princeton, NJ, 1970, xiv+290 pp. | MR | MR | Zbl | Zbl

[4] M. F. Timan, “The best approximation of a function and linear methods for the summation of Fourier series”, Amer. Math. Soc. Transl. Ser. 2, 77, Amer. Math. Soc., Providence, RI, 1968, 127–147 | DOI | MR | Zbl

[5] R. M. Trigub, “Absolute convergence of Fourier integrals, summability of Fourier series, and polynomial approximation of functions on the torus”, Math. USSR-Izv., 17:3 (1981), 567–593 | DOI | MR | Zbl

[6] E. Liflyand, S. Samko, R. Trigub, “The Wiener algebra of absolutely convergent Fourier integrals: an overview”, Anal. Math. Phys., 2:1 (2012), 1–68 | DOI | MR | Zbl

[7] R. M. Trigub, “Fourier transformation of quasiconvex functions and functions of the class $V^*$”, J. Math. Sci. (N.Y.), 204:3 (2015), 369–378 | DOI | MR | Zbl

[8] A. Zygmund, “Smooth functions”, Duke Math. J., 12:1 (1945), 47–76 | DOI | MR | Zbl

[9] N. K. Bari, S. B. Stechkin, “Nailuchshie priblizheniya i differentsialnye svoistva dvukh sopryazhennykh funktsii”, Tr. MMO, 5, GITTL, M., 1956, 483–522 | MR | Zbl

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

[11] R. A. DeVore, G. G. Lorentz, Constructive approximation, Grundlehren Math. Wiss., 303, Springer-Verlag, Berlin, 1993, x+449 pp. | MR | Zbl

[12] S. A. Telyakovskii, “O rabotakh S. B. Stechkina po priblizheniyu periodicheskikh funktsii polinomami”, Fundament. i prikl. matem., 3:4 (1997), 1059–1068 | MR | Zbl

[13] V. V. Zhuk, G. I. Natanson, “S. N. Bernstein and direct and converse theorems of constructive function theory”, Proceedings of the St. Petersburg Mathematical Society, v. VIII, Amer. Math. Soc. Transl. Ser. 2, 205, Amer. Math. Soc., Providence, RI, 2002, 59–82 | MR | Zbl

[14] M. K. Potapov, B. V. Simonov, “Moduli gladkosti polozhitelnykh poryadkov funktsii iz prostranstv $L_p$, $1 \le p \le +\infty$”, Sovremennye problemy matematiki i mekhaniki, 7, no. 1, Izd-vo mekh.-matem. f-ta MGU, M., 2011, 100–109

[15] K. V. Runovski, “A direct theorem of approximation theory for a general modulus of smoothness”, Math. Notes, 95:6 (2014), 833–842 | DOI | DOI | MR | Zbl

[16] M. K. Potapov, B. V. Simonov, S. Yu. Tikhonov, Drobnye moduli gladkosti, MAKS PRESS, M., 2016, 338 pp.

[17] B. Szökefalvi-Nagy, “Über gewisse Extremalfragen bei transformierten trigonometrischen Entwicklungen. I. Periodischer Fall”, Ber. Verh. Sächs. Akad. Leipzig, 90 (1938), 103–134 | Zbl

[18] S. B. Stechkin, “O nailuchshem priblizhenii sopryazhennykh funktsii trigonometricheskimi polinomami”, Izv. AN SSSR. Ser. matem., 20:2 (1956), 197–206 | MR | Zbl

[19] A. Zygmund, Trigonometric series, v. I, II, 2nd ed., Cambridge Univ. Press, New York, 1959, xii+383 pp., vii+354 pp. | MR | MR | Zbl | Zbl

[20] M. K. Potapov, “Interconnection between certain classes of functions”, Math. Notes, 2:4 (1967), 706–714 | DOI | MR | Zbl

[21] P. L. Butzer, R. J. Nessel, Fourier analysis and approximation, v. 1, Pure Appl. Math., 40, Academic Press, New-York–London; Birkhäuser Verlag, Basel, 1971, xvi+553 pp. | DOI | MR | Zbl

[22] J. Boman, H. S. Shapiro, “Comparison theorems for a generalized modulus of continuity”, Ark. Mat., 9:1-2 (1971), 91–116 | DOI | MR | Zbl

[23] J. Boman, “Equivalence of generalized moduli of continuity”, Ark. Mat., 18:1-2 (1980), 73–100 | DOI | MR | Zbl

[24] A. I. Stepanets, Classification and approximation of periodic functions, Math. Appl., 333, Kluwer Acad. Publ., Dordrecht, 1995, x+360 pp. | DOI | MR | MR | Zbl | Zbl

[25] H. Triebel, Higher analysis, Hochschulbücher fur Math., Johann Ambrosius Barth Verlag GmbH, Leipzig, 1992, 473 pp. | MR | Zbl

[26] M. K. Potapov, B. V. Simonov, “On the interrelation of the generalized Besov–Nikol'skiĭ and Weyl–Nikol'skiĭ classes of functions”, Anal. Math., 22:4 (1996), 299–316 | DOI | MR | Zbl

[27] A. I. Stepanets, Methods of approximation theory, VSP, Leiden, 2005, xviii+919 pp. | DOI | MR | Zbl

[28] B. V. Simonov, S. Yu. Tikhonov, “On embeddings of function classes defined by constructive characteristics”, Approximation and probability, Banach Center Publ., 72, Polish Acad. Sci. Inst. Math., Warsaw, 2006, 285–307 | DOI | MR | Zbl

[29] B. V. Simonov, S. Yu. Tikhonov, “Embedding theorems in constructive approximation”, Sb. Math., 199:9 (2008), 1367–1407 | DOI | DOI | MR | Zbl

[30] K. V. Runovskii, Priblizhenie semeistvami lineinykh polinomialnykh operatorov, Dis. $\dots$ dokt. fiz.-matem. nauk, MGU, M., 2010, 236 pp.

[31] K. Runovski, H.-J. Schmeisser, “Smoothness and function spaces generated by homogeneous multipliers”, J. Funct. Spaces Appl., 2012 (2012), 643135, 22 pp. | DOI | MR | Zbl

[32] K. V. Runovski, H.-J. Schmeisser, “Moduli of smoothness related to fractional Riesz-derivatives”, Z. Anal. Anwend., 34:1 (2015), 109–125 | DOI | MR | Zbl

[33] K. V. Runovskii, “Trigonometric polynomial approximation, $K$-functionals and generalized moduli of smoothness”, Sb. Math., 208:2 (2017), 237–254 | DOI | DOI | MR | Zbl

[34] K. V. Runovskii, “Generalized smoothness and approximation of periodic functions in the spaces $L_p$, $1

+\infty$”, Math. Notes, 106:3 (2019), 412–422 | DOI | DOI | MR | Zbl

[35] S. B. Stečkin, “The approximation of periodic functions by Fejér sums”, Amer. Math. Soc. Transl. Ser. 2, 28, Amer. Math. Soc., Providence, RI, 1963, 269–282 | DOI | MR | Zbl

[36] M. F. Timan, V. G. Ponomarenko, “The approximation of periodic functions of two variables by sums of Marcinkiewicz type”, Soviet Math. (Iz. VUZ), 19:9 (1975), 49–56 | MR | Zbl

[37] K. I. Oskolkov, “Lebesgue's inequality in a uniform metric and on a set of full measure”, Math. Notes, 18:4 (1975), 895–902 | DOI | MR | Zbl

[38] K. Runovski, H.-J. Schmeisser, “On the convergence of Fourier means and interpolation means”, J. Comput. Anal. Appl., 6:3 (2004), 211–227 | MR | Zbl

[39] R. M. Trigub, E. S. Bellinsky, Fourier analysis and approximation of functions, Kluwer Acad. Publ., Dordrecht, 2004, xiv+585 pp. | DOI | MR | Zbl

[40] K. Runovski, H.-J. Schmeisser, “On approximation methods generated by Bochner–Riesz kernels”, J. Fourier Anal. Appl., 14:1 (2008), 16–38 | DOI | MR | Zbl

[41] R. M. Trigub, “Exast order of approximation of periodic functions by linear polynomial operators”, East J. Approx., 15:1 (2009), 25–50 | MR | Zbl

[42] V. A. Gerasimenko, Yu. S. Kolomoitsev, “Ob ekvivalentnosti $K$-funktsionalov i approksimatsionnykh metodov, porozhdennykh obobschennymi yadrami Bokhnera–Rissa”, Vestn. Khark. un-ta. Ser. matem., prikl. matem. i mekh., 922:61 (2010), 56–64 | Zbl

[43] V. V. Zhuk, “Estimates of best approximations of a periodic function by linear combinations of values of the function and its primitives”, J. Math. Sci. (N.Y.), 193:1 (2013), 89–99 | DOI | MR | Zbl

[44] K. V. Runovski, “Approximation by Fourier means and generalized moduli of smoothness”, Math. Notes, 99:4 (2016), 564–575 | DOI | DOI | MR | Zbl

[45] S. B. Stechkin, “Obobschenie nekotorykh neravenstv S. N. Bernshteina”, Dokl. AN SSSR, 60:9 (1948), 1511–1514 | MR | Zbl

[46] S. N. Bernshtein, Sobranie sochinenii, v. 1, Izd-vo AN SSSR, M., 1952, 581 pp. | MR | Zbl

[47] V. V. Arestov, “On inequalities of S. N. Bernstein for algebraic and trigonometric polynomials”, Soviet Math. Dokl., 20 (1979), 600–603 | MR | Zbl

[48] V. E. Maiorov, “Bernshtein–Nikol'skii inequalities and estimates of the norms of Dirichlet kernels for trigonometric polynomials over arbitrary harmonics”, Math. Notes, 47:6 (1990), 565–569 | DOI | MR | Zbl

[49] A. I. Kozko, “Fractional derivatives and inequalities for trigonometric polynomials in spaces with asymmetric norms”, Izv. Math., 62:6 (1998), 1189–1206 | DOI | DOI | MR | Zbl