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Approximation of functions by the Fourier-Walsh sums. (English. Russian original) Zbl 0626.42003
Math. Notes 41, 15-22 (1987); translation from Mat. Zametki 41, No. 1, 23-33 (1987).
The author proves the following theorem: Let X be the space \(L^ 1[0,1]\) or \(L^{\infty}[0,1]\). Then for every natural number \[ \sup_{f\in X_{\epsilon}}\| f-S_ N(f)\|_ x\epsilon_ N+\sum^{N}_{n=1}\epsilon_{N+n}\int^{1/n}_{1/(n+1)}| D_ N(x)| dx, \] where \(D_ N(x)\) is the Dirichlet-Walsh kernel. In the trigonometric case equivalent results were obtained by the author and K. I. Oskolkov.
Reviewer: R.Gajewski
MSC:
42A10 Trigonometric approximation
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
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References:
[1] J. L. Walsh, ?A closed set of normal orthogonal functions,? Am. J. Math.,45, 5-24 (1923). · JFM 49.0293.03 · doi:10.2307/2387224
[2] R. E. A. C. Paley, ?A remarkable system of orthogonal functions,? Proc. London Math. Soc.,34, 241-279 (1932). · Zbl 0005.24806 · doi:10.1112/plms/s2-34.1.241
[3] N. Ya. Vilenkin, ?A class of complete orthogonal systems,? Izv. Akad. Nauk SSSR, Ser. Mat.2, 363-400 (1947).
[4] S. B. Stechkin, ?On the approximation of periodic functions by the Fejer sums,? Trudy MI AN SSSR,62, 48-60 (1961).
[5] V. E. Geit, ?Exactness of some inequalities in the approximation theory,? Mat. Zametki,10, No. 5, 571-582 (1971). · Zbl 0235.42001
[6] K. I. Oskolkov, ?On the Lebesgue inequality in the uniform metric and on a set of full measure,? Mat. Zametki,18, No. 4, 515-526 (1975). · Zbl 0339.42001
[7] S. B. Stechkin, ?On the approximation of periodic functions by de la Vallee Poussin sums,? Analysis Mathematica,4, 61-74 (1978). · Zbl 0393.41009 · doi:10.1007/BF01904859
[8] V. Damen, ?On the best approximation and de la Vallee Poussin sums,? Mat. Zametki,23, No. 5, 671-683 (1978). · Zbl 0385.42001
[9] K. I. Oskolov, ?On the Lebesgue inequality in mean,? Mat. Zametki,25, No. 4, 551-555 (1979).
[10] S. P. Baiborodov, ?Function approximation by de la Vallee Poussin sums,? Mat. Zametki,27, No. 1, 33-48 (1980).
[11] S. P. Baiborodov, ?An approximation of multivariable functions by the orthogonal de la Vallee Poussin sums,? Mat. Zametki,29, No. 5, 711-730 (1981).
[12] Wang Kunyang, ?The estimation of the multiple de la Vallee Poussin square remainder,? Chinese Ann. Math.,3, No. 6, 789-802 (1982).
[13] S. P. Baiborodov, ?The Lebesgue constants and the function approximation by the orthogonal Fourier sums in LP(Tm),? Mat. Zametki,34, No. 1, 77-90 (1983).
[14] S. P. Baiborodov, ?The approximation of functions of many variables by the Fejer sums,? Mat. Zametki,36, No. 1, 123-136 (1984).
[15] N. A. Il’yasov, ?The approximation of periodic functions by the Zygmund conjugates,? Mat. Zametki,39, No. 3, 367-382 (1986).
[16] I. I. Sharapudinov, ?On the best approximation and the Fourier-Jacobi sums,? Mat. Zametki,34, No. 5, 651-661 (1983). · Zbl 0532.42010
[17] B. S. Kashin and A. A. Saakyan, Orthogonal Series [in Russian], Nauka, Moscow (1984).
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