The exact value of the upper bound of the approximation error by the construction of the type "angle" built using the Fourier-Haar partial sums is obtained on the function class of two variables in the uniform metric.

Periodical:

International Journal of Advanced Research in Mathematics (Volume 5)

Pages:

23-31

Citation:

A. Shchitov, "On Approximation of the Continuous Functions of Two Variables by the Fourier-Haar ”Angle”", International Journal of Advanced Research in Mathematics, Vol. 5, pp. 23-31, 2016

Online since:

June 2016

Authors:

Keywords:

Distribution:

Open Access

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Creative Commons Attribution 4.0 International License

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. Yu. Galkina, On the Fourier-Haar coefficients of functions of bounded variation, Mathematical Notes. 51: 1 (1992) 27-36.

[7] S. Yu. Galkina, On the Fourier-Haar Coefficients of Functions of Several Variables with Bounded Vitali Variation, Mathematical Notes. 70: 5 (2001) 733-743.

[8] 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.

[9] A.N. Shchitov, Sharp 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).

[10] 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.

[11] M.G. Plotnikov, Coefficients of convergent multiple Haar series, Russian Mathematics. 56: 1 (2012) 61-65.

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

[13] 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.

[14] 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).

[15] 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).

[16] 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).

[17] 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.

[18] 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).

[19] 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).

[20] 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).

[21] 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).

[22] 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).

[23] 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).

[24] M.K. Potapov, The study of some classes of functions by the help of the approximation by the angle, Proceedings of the Steklov Institute of Mathematics. 117 (1972) 256-291. (in Russian).

[25] M.K. Potapov, On the approximation by the angle, Proceedings of the Conference of Constructive Theory of Functions. 1969. (in Russian).

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

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

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

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

[30] O.D. Gabisonia, On the value of the best approximation of functions of several variables by polynomials in the Haar system, Izv. Vyssh. Uchebn. Zaved., Mat. 1 (1968) 39-46. (in Rusian).

Cited By:

[1] A. 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, p. 42, 2016

DOI: https://doi.org/10.18052/www.scipress.com/IJARM.6.42