In the article are studied some problems of approximation theory in the spaces *S*^{p} (1 ≤ *p* < ∞) introduced by A.I. Stepanets. It is obtained the exact values of extremal characteristics of a special form which connect the values of best polynomial approximations of functions *e*_{n}_{-1}(*f*)*S*^{p} with expressions which contain modules of continuity of functions *f*(*x*) є *S*^{p}. We have obtained the asymptotically sharp inequalities of Jackson type that connect the best polynomial approximations *e*_{n}_{-1}(*f*)*S*^{p} with modules of continuity of functions *f*(*x*) є *S*^{p} (1 ≤ *p* < ∞). Exact values of Kolmogorov, linear, Bernstein, Gelfand and projection n-widths in the spaces *S*^{p} are obtained for some classes of functions *f*(*x*) є *S*^{p}. The upper bound of the Fourier coefficients are found for some classes of functions.

Periodical:

International Journal of Advanced Research in Mathematics (Volume 7)

Pages:

19-32

Citation:

A. N. Shchitov "On Best Polynomial Approximations in the Spaces *S*^{p} and Widths of Some Classes of Functions", International Journal of Advanced Research in Mathematics, Vol. 7, pp. 19-32, 2016

Online since:

Dec 2016

Authors:

Keywords:

Distribution:

Open Access

This work is licensed under a

Creative Commons Attribution 4.0 International License

References:

[1] N.I. Chernykh, Best approximation of periodic functions by trigonometric polynomials in L2, Mathematical notes of the Academy of Sciences of the USSR. 2(5) (1967) 803-808.

DOI: 10.1007/bf01093942[2] N.I. Chernykh, On Jackson's inequality in L2, Proceedings of the Steklov Institute of Mathematics. 88 (1967) 75-78. (in Russian).

[3] L.V. Taikov, Inequalities containing best approximations and the modulus of continuity of functions in L2, Mathematical notes of the Academy of Sciences of the USSR. 20(3) (1976) 797-800.

DOI: 10.1007/bf01097254[4] L.V. Taikov, Structural and constructive characteristics of functions in L2, Mathematical notes of the Academy of Sciences of the USSR. 25(2) (1979) 113-116.

DOI: 10.1007/bf01142721[5] A.A. Ligun Some inequalities between best approximations and moduli of continuity in an L2 space, Mathematical notes of the Academy of Sciences of the USSR. 24(6) (1978) 917-921.

DOI: 10.1007/bf01140019[6] V.V. Shalaev, Widths in L2 of classes of differentiable functions, defined by higher-order moduli of continuity, Ukrainian Mathematical Journal. 43(1) (1991) 104-107.

DOI: 10.1007/bf01066914[7] Kh. Yussef, On the best approximation of the functions and values of widths of classes of functions in L2, in: Collection of Scientific Works Application of Functional Analysis to the Theory of Approximations, Kalinin, USSR, 1988, pp.100-114.

[8] S.B. Vakarchuk, On Best Polynomial Approximations in L2, Mathematical Notes. 70(3) (2001) 300-310.

[9] S.B. Vakarchuk, Generalized smoothness characteristics in Jackson-type inequalities and widths of classes of functions in L2, Mathematical Notes. 98(3) (2015) 572-588.

DOI: 10.1134/s0001434615090254[10] S.B. Vakarchuk, Jackson-type inequalities with generalized modulus of continuity and exact values of the n-widths for the classes of (Ψ, β)-differentiable functions in L2, I, Ukrainian Mathematical Journal. 68(6) (2016) 823-848.

DOI: 10.1007/s11253-016-1260-z[11] S.B. Vakarchuk, V.I. Zabutnaya, Inequalities between best polynomial approximations and some smoothness characteristics in the space L2 and widths of classes of functions, Mathematical Notes. 99(1) (2016) 222-242.

DOI: 10.1134/s0001434616010259[12] S.B. Vakarchuk, V.I. Zabutnaya, On the best polynomial approximation in the space L2 and widths of some classes of functions, Ukrainian Mathematical Journal. 64(8) (2012) 1168-1176.

DOI: 10.1007/s11253-013-0707-8[13] S.B. Vakarchuk, V.I. Zabutnaya, Jackson-Stechkin type inequalities for special moduli of continuity and widths of function classes in the space L2, Mathematical Notes. 92(3) (2012) 458-472.

DOI: 10.1134/s0001434612090180[14] S.N. Vasilyev, Exact of Jackson-Stechkin type inequalities in L2 with a modulus of continuity, generated by an arbitrary finite-difference operator with constant coefficients, Dokl. Russian Academy of Sciences. 385(1) (2002) 11-14. (in Russian).

[15] M.S. Shabozov, S.B. Vakarchuk, On the best approximation of periodic functions by trigonometric polynomials and the exact values of widths of function classes in L2, Analysis Mathematica. 38(2) (2012) 147-159.

[16] A.G. Babenko, On the Jackson-Stechkin inequality for the best L2-approximations of functions by trigonometric polynomials, Proc. Steklov Inst. Math., Suppl. 1 (2001) 30-47.

[17] M. Sh. Shabozov, G.A. Yusupov, Exact constants in Jackson-type inequalities and exact values of the widths of some classes of functions in L2, Sib. Math. J. 52(6) (2011) 1124-1136.

DOI: 10.1134/s0037446611060176[18] S.B. Vakarchuk, A.N. Shchitov, Best polynomial approximations in L2 and widths of some classes of functions, Ukrainian Mathematical Journal. 56(11) (2004) 1738-1747.

DOI: 10.1007/s11253-005-0148-0[19] A.I. Stepanets, Approximation Characteristics of Spaces Sp ϕ, Ukrainian Mathematical Journal. 53(3) (2001) 446-475.

[20] A.I. Stepanets, Approximation Characteristics of the Spaces Sϕ p in Different Metrics, Ukrainian Mathematical Journal. 53(8) (2001) 1340-1374.

[21] A.S. Serdyuk, Widths in the space Sp of classes of functions defined by moduli of continuity, in: Proceedings of the Institute of Mathematics of the Ukrainian National Academy of Sciences Extremal Problems of the Theory of Functions and Related Problems, Vol. 46, Kyiv, Ukraine, 2003, pp.229-248.

[22] V.R. Voitsekhivs'kyi, Jackson-Type Inequalities in the Space Sp, Ukrainian Mathematical Journal. 55(9) (2003) 1410-1422.

[23] S.B. Vakarchuk, Jackson-type inequalities and exact values of widths of classes of functions in the spaces Sp, 1 6 p < ∞, Ukrainian Mathematical Journal. 56(5) (2004) 718-729.

DOI: 10.1007/s11253-005-0070-5[24] S.B. Vakarchuk, A.N. Shchitov, On some extremal problems in the theory of approximation of functions in the spaces Sp, 1 6 p < ∞, Ukrainian Mathematical Journal. 58(3) (2006) 340-356.

DOI: 10.1007/s11253-006-0070-0[25] A. Kolmogoroff, Über die besste Annäherung von Funktionen einer gegebenen Funktionklassen, Ann. of Math. 37 (1936) 107-110.

DOI: 10.2307/1968691[26] A.I. Stepanets, A.S. Serdyuk, Direct and inverse theorems in the theory of approximation of functions in the space Sp, Ukrainian Mathematical Journal. 54(1) (2002) 126-148.

[27] N. Ainulloev, Values of widths of certain classes of differentiable functions in L2, Dokl. Akad. Nauk Tadzh. SSR. 27(8) (1984) 415-418.

[28] N. Ainulloev, The best approximation for some classes of differentiable functions in L2, in: Application of Functional Analysis in Approximation Theory, Kalinin, USSR, 1986, pp.3-10. (in Russian).

[29] V.A. Andrienko, Embedding theorems for functions of one variable, Journal of Soviet Mathematics. 6(1) (1973) 764-804.

DOI: 10.1007/bf01236364[30] V.R. Voitsekhivs'kyi, Widths of certain classes from the space Sp, in: Proceedings of the Institute of Mathematics of the Ukrainian National Academy of Sciences Extremal Problems of the Theory of Functions and Related Problems, Vol. 46, Kyiv, Ukraine, 2003, pp.17-26.

[31] M.G. Esmaganbetov, Widths of classes from L2[0, 2π] and minimization of exact constants in Jackson-type inequalities, Mathematical Notes. 65(6) (1999) 689-693.

DOI: 10.1007/bf02675582[32] K.G. Ivanov, New Estimates of Errors of Quadrature Formulae, Formulae of Numerical Differention and Interpolation, Analysis Math. 6(4) (1980) 281-303.

DOI: 10.1007/bf02053634[33] K.G. Ivanov, On a New Characteristic of Functions. I, Serdica. 8(3) (1982) 262-279.

[34] N.P. Korneichuk, Splines in approximation theory, Moskov, USSR, 1984. (in Russian).

[35] A.I. Stepanets, Classification and approximation of periodic functions, Naukova Dumka, Kiev, Ukrainian SSR, 1987. (in Russian).

[36] V.M. Tikhomirov, Theory of approximations, Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fund. Naprav. 14 (1987) 103-260. (in Russian).

[37] Kh. Yussef, Widths of classes of functions in the space L2(0, 2π), in: Collection of Scientific Works Application of Functional Analysis to the Theory of Approximations, Tver University, Tver, USSR, 1990, pp.167-175. (in Russian).