, are investigated for periodic functions of an arbitrary number of variables in terms of the generalized modulus of smoothness generated by a periodic generator which, near the origin, is assumed to be close in a certain sense to some homogeneous function of positive order. Direct and inverse theorems (Jackson- and Bernstein-type estimates) are proved; conditions on the generators are obtained under which the approximation error by an FLPO is equivalent to an appropriate modulus of smoothness. These problems are solved by going over from the modulus to an equivalent $K$-functional. The general results obtained are applied to classical objects in the theory of approximation and smoothness. In particular, they are applied to the methods of approximation generated by Fejér, Riesz and Bochner-Riesz kernels, and also to the moduli of smoothness and $K$-functionals corresponding to the conventional, Weyl and Riesz derivatives and to the Laplace operator. Bibliography: 24 titles.
@article{SM_2017_208_2_a3,
author = {K. V. Runovski},
title = {Trigonometric polynomial approximation, $K$-functionals and generalized moduli of smoothness},
journal = {Sbornik. Mathematics},
pages = {237--254},
year = {2017},
volume = {208},
number = {2},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SM_2017_208_2_a3/}
}
K. V. Runovski. Trigonometric polynomial approximation, $K$-functionals and generalized moduli of smoothness. Sbornik. Mathematics, Tome 208 (2017) no. 2, pp. 237-254. http://geodesic.mathdoc.fr/item/SM_2017_208_2_a3/
[1] K. V. Runovskiĭ, “On families of linear polynomial operators in $L_p$-spaces, $0
1$”, Russian Acad. Sci. Sb. Math., 78:1 (1994), 165–173 | DOI | MR | Zbl[2] K. V. Runovskiĭ, “On approximation by families of linear polynomial operators in $L_p$-spaces, $0
1$”, Russian Acad. Sci. Sb. Math., 82:2 (1995), 441–459 | DOI | MR | Zbl[3] V. I. Rukasov, K. V. Runovski, H.-J. Schmeisser, “On convergence of families of linear polynomial operators”, Funct. Approx. Comment. Math., 41:1 (2009), 41–54 | DOI | MR | Zbl
[4] V. Rukasov, K. Runovski, H.-J. Schmeisser, “Approximation by families of linear trigonometric polynomial operators and smoothness properties of functions”, Math. Nachr., 284:11-12 (2011), 1523–1537 | DOI | MR | Zbl
[5] K. V. Runovskii, Priblizhenie semeistvami lineinykh polinomialnykh operatorov, Diss. $\dots$ dokt. fiz.-matem. nauk, MGU, M., 2010, 236 pp.
[6] K. Runovski, H.-J. Schmeisser, “On convergence of families of linear polynomial operators generated by matrices of multipliers”, Eurasian Math. J., 1:3 (2010), 112–133 | MR | Zbl
[7] 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
[8] K. Runovski, H.-J. Schmeisser, “On families of linear polynomial operators generated by Riesz kernels”, Eurasian Math. J., 1:4 (2010), 124–139 | MR | Zbl
[9] V. H. Hristov, K. G. Ivanov, “Realizations of $K$-functionals on subsets and constrained approximation”, Math. Balkanica (N.S.), 4:3 (1990), 236–257 | MR | Zbl
[10] 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
[11] Z. Ditzian, V. Hristov, K. Ivanov, “Moduli of smoothness and $K$-functionals in $L_p$, $0
1$”, Constr. Approx., 11:1 (1995), 67–83 | DOI | MR | Zbl[12] Z. Ditzian, “On Fejer and Bochner–Riesz means”, J. Fourier Anal. Appl., 11:4 (2005), 489–496 | DOI | MR | Zbl
[13] R. A. DeVore, G. G. Lorentz, Constructive approximation, Grundlehren Math. Wiss., 303, Springer-Verlag, Berlin, 1993, x+449 pp. | MR | Zbl
[14] M. K. Potapov, B. V. Simonov, “Moduli gladkosti polozhitelnykh poryadkov funktsii iz prostranstv $L_p$, $1 \le p \le \infty$”, K 190-letiyu P. L. Chebysheva, Sovremennye problemy matematiki i mekhaniki, 7, no. 1, Izd-vo Mosk. un-ta, M., 2011, 100–109
[15] Z. Ditzian, “A measure of smoothness related to the Laplacian”, Trans. Amer. Math. Soc., 326:1 (1991), 407–422 | DOI | MR | Zbl
[16] Z. Ditzian, K. Runovskii, “Realization and smoothness related to the Laplacian”, Acta Math. Hungar., 93:3 (2001), 189–223 | DOI | MR | Zbl
[17] J. Boman, H. S. Shapiro, “Comparison theorems for a generalized modulus of continuity”, Ark. Mat., 9 (1971), 91–116 | DOI | MR | Zbl
[18] J. Boman, “Equivalence of generalized moduli of continuity”, Ark. Mat., 18:1 (1980), 73–100 | DOI | MR | Zbl
[19] K. Runovski, H.-J. Schmeisser, “On some extensions of Bernstein's inequality for trigonometric polynomials”, Funct. Approx. Comment. Math., 29 (2001), 125–142 | MR
[20] K. Runovski, H.-J. Schmeisser, “Inequalities of Calderón–Zygmund type for trigonometric polynomials”, Georgian Math. J., 8:1 (2001), 165–179 | MR | Zbl
[21] K. Runovski, H.-J. Schmeisser, “General moduli of smoothness and approximation by families of linear polynomial operators”, New perspectives on approximation and sampling theory, Appl. Numer. Harmon. Anal., Birkhäuser/Springer, Cham, 2014, 269–298 | DOI | MR | Zbl
[22] K. Runovski, H.-J. Schmeisser, On modulus of continuity related to Riesz derivative, Preprint, Friedrich-Schiller-Universität, Jena, 2011, 12 pp.
[23] 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
[24] K. Runovski, H.-J. Schmeisser, “Moduli of smoothness related to the Laplace-operator”, J. Fourier Anal. Appl., 21:3 (2015), 449–471 | DOI | MR | Zbl