Subscribe

Subscribe to our Newsletter and get informed about new publication regulary and special discounts for subscribers!

IJARM > Volume 6 > Best One-Sided Approximation of Some Classes of...
< Back to Volume

Best One-Sided Approximation of Some Classes of Functions of Several Variables by Haar Polynomials

Full Text PDF

Abstract:

Exact values of the best one-sided approximation by Haar polynomials have been obtained in the integral and uniform metrics for some classes of the functions of several variables defined using modulus of continuity ω(f;t) and ωρi (f;δ).

Info:

Periodical:
International Journal of Advanced Research in Mathematics (Volume 6)
Pages:
42-50
Citation:
A. N. Shchitov "Best One-Sided Approximation of Some Classes of Functions of Several Variables by Haar Polynomials", International Journal of Advanced Research in Mathematics, Vol. 6, pp. 42-50, 2016
Online since:
September 2016
Export:
Distribution:
References:

[1] A. Haar, Zur Theorie der orthogonalen Funktionensysteme, Inaugural-dissertation, GeorgAugust-Universitat, Göttingen, (1909).

[2] P.L. Ulyanov, Series in Haar system, Matem. sbornik. 63(1) (1964) 356-391. (in Russian).

[3] Z. Ciesielski, On Haar functions and on the Schauder Basis of the Space C⟨0, 1⟩, Bulletin de L'academie Polonaise Des Sciences. Serie des sci. math., astr. et phys. 7(4) (1959) 227-232.

[4] B.I. Golubov, On Fourier series of continuous functions with respect to a Haar system, Izv. Akad. Nauk SSSR. Ser. Mat. 28 (1964) 1271-1296. (in Russian).

[5] I.I. Sharapudinov, Approximation of functions in variable-exponent Lebesgue and Sobolev spaces by finite Fourier-Haar series, Sb. Math. 205(2) (2014) 291-306.

[6] S.S. Volosivets, Approximation of functions of bounded p-variation by means of polynomials of the Haar and Walsh systems, Mathematical Notes. 53(6) (1993) 569-575.

[7] A.N. Shchitov, Exact estimates of the Fourier-Haar coefficients of some classes of functions of several variables, Zb. Pr. Inst. Mat. NAN Ukr. 1(1) (2004) 413-427. (in Russian).

[8] A.N. Shchitov, The exact estimates of Fourier-Haar coefficients of functions of bounded variation, International Journal of Advanced Research in Mathematics. 4 (2016) 14-22.

[9] B.I. Golubov, Series with respect to the Haar system, J. Soviet Math. 1(6) (1973) 704-726.

[10] N.P. Khoroshko, On the best approximation in the metric of L to certain classes of functions by Haar-system polynomials, Mathematical notes of the Academy of Sciences of the USSR. 6(1) (1969) 487-491.

[11] N.P. Khoroshko, Uniform approximation in classes of continuous functions by polynomials in the Haar system, Ukrainian Math. J. 22(5) (1971) 611-618. (in Russian).

[12] S.B. Vakarchuk, A.N. Shchitov, On the best approximation of functions of bounded p-variation by Haar polynomials, Bulletin of the Dnipropetrovsk University. Mathematics. 11 (2004) 28-34. (in Russian).

[13] S.B. Vakarchuk, A.N. Shchitov, Estimates for the error of approximation of functions in L1 p by polynomials and partial sums of series in the Haar and Faber-Schauder systems, Izvestiya Mathematics. 79(2) (2015) 257-287.

[14] A.R. Abdulgamidov, On some properties of the Fourier-Haar series of the functions of two variables, Proceedings of the postraduate papers of the Kazan state university: Mathematics, Mechanics, Physics. (1968) 5-33. (in Russian).

[15] N.D. Rishchenko, On the best approximation of the function of the several variables by the step functions, Proccedings of the Dnipropetrovsk state university. (1969) 56-59. (in Russian).

[16] M.N. Ochirov, On approximation of functions of two variables by Fourier-Haar partial sums, Proccedings of the Kazan state university: Function analyse and function theory. 8 (1971) 142- 145. (in Russian).

[17] L.G. Khomutenko, Uniform approximation of functions of two variables by polynomials in the Haar system, Proccedings of the Dnipropetrovsk state university. (1973) 85-87. (in Russian).

[18] P.V. Zaderey, N.N. Zaderey, Uniform approximations by polynomials in the Haar system on classes of continuous functions of several variables, Zb. Pr. Inst. Mat. NAN USSR. (1984) 60- 66. (in Russian).

[19] S.B. Vakarchuk, A.N. Shchitov, Uniform approximation of some classes of functions of several variables by polynomials constructed by Haar system and partial sums of the Fourier-Haar series, Zb. Pr. Inst. Mat. NAN Ukr. 1(1) (2004).

[20] A.N. Shchitov, On approximation of the continuous functions of two variables by the FourierHaar polynomials, International Journal of Advanced Research in Mathematics. 5 (2016) 23-31.

[21] G. Freud, Uber cinseitige Approximation durch Polynome, I, Acta Sci. Math. (Szeged). 16(2) (1955) 12-28.

[22] T. Ganelius, On the sided approximation by trigonometric polynomials, Math. Scand. 4 (1956) 247-258.

[23] N.P. Korneichuk, A.A. Ligun, V.G. Doronin, Approximation with constraints, Naukova dumka, Kiev, Ukrainian SSR, 1982. (in Russian).

[24] V.G. Doronin, A.A. Ligun, On the study of the best one-sided approximation of some function classes of the continuous functions, Proccedings of the Dnipropetrovsk state university. (1974) 42-49. (in Russian).

[25] S.B. Vakarchuk, A.N. Shchitov, Estimates of the best one-sided approximation of some classes of functions by polynomials constructed on Haar system, Zb. Pr. Inst. Mat. NAN Ukr. 4 (2007) 7-22. (in Russian).

[26] N.P. Korneichuk, Exact constants in approximation theory, Encyclopedia of Mathematics and its Applications, 38, Cambridge University Press, Cambridge, (1991).

[27] G. Alexits, Convergence problems of orthogonal series, Akademia Kiado, Budapest, Hungary, (1961).

[28] B.S. Kashin, A.A. Saakyan, Orthogonal series, Transl. Math. Monogr., 75, Amer. Math. Soc., Providence, RI, (1989).

[29] S.S. Kaczmarz, H. Steinhaus, Theorie der Orthogonalreihen, Monografje Matematyczne, VI, Chelsea Publishing Company, New York, (1951).

[30] I.M. Sobol, Multidimensional quadrature formulas and functions Haar, Nauka, Moscow, USSR, 1969. (in Russian).

[31] N.P. Korneichuk, Extremal problems in the theory of approximation, Nauka, Moskow, USSR, 1976. (in Russian).

[32] N.P. Korneichuk, Splines in approximation theory, Nauka, Moskow, USSR, 1984. (in Russian).

Show More Hide