Abstract
It is well known that if \({f \in L_{2}(0,1)}\) is an arbitrary function (\({{f(x) \nsim 0}, x \in [0,1]}\)) then its Fourier coefficients with respect to general orthonormal systems (ONS) may belong only to \({\ell_2}\). Thus in the general case it is impossible to estimate these coefficients by moduli of continuity or moduli of smoothness of the given functions.
In the present paper conditions are found which should be satisfied by ONS so that the coefficients of some classes of functions can be estimated by modulus of continuity or modulus of smoothness of these functions.
Key words and phrases
general orthonormal system Fourier coefficient modulus of continuity modulus of smoothnessMathematics Subject Classification
42C10Preview
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References
- 1.N. K. Bary, Trigonometric Series, GIFM (Moscow, 1961) (in Russian)Google Scholar
- 2.Fine, N.J.: On the Walsh functions. Trans. Amer. Math. Soc. 65, 372–414 (1949)MathSciNetCrossRefzbMATHGoogle Scholar
- 3.L. Gogoladze and V. Tsagareishvili, Fourier coefficients of continuous functions, Mat. Zametki, 91 (2012), 691–703 (in Russian); translation in Math. Notes, 91 (2012), 645–656.Google Scholar
- 4.S. Kaczmarz and G. Steinhaus, Theory of Orthogonal Series, Gosudarstv. Izdat. Fiz.-Mat. Lit. (Moscow, 1958) (in Russian)Google Scholar
- 5.B. S. Kashin and A. A. Saakyan, Orthogonal Series, 2nd ed., Izdat. Nauchno-Issledovatel'skogo Aktuarno-Finansovogo Tsentra (AFTs) (Moscow, 1999) (in Russian)Google Scholar
- 6.McLaughlin, J.R.: Integrated orthonormal series. Pacific J. Math. 42, 469–475 (1972)MathSciNetCrossRefzbMATHGoogle Scholar
- 7.A. M. Olevskiĭ, Orthogonal series in terms of complete systems, Mat. Sb. (N.S.), 58 (1962), 707–748 (in Russian)Google Scholar
- 8.Tsagareishvili, V.: On the Fourier coefficients for general orthonormal systems. Proc. A. Razmadze Math. Inst. 124, 131–150 (2000)MathSciNetzbMATHGoogle Scholar
- 9.V. Tsagareishvili, On the variation of the Fourier-Haar coefficients, Mat. Sb., 195: 143–160 (in Russian); translation in Sb. Math. 195(2004), 441–457 (2004)Google Scholar
- 10.Tsagareishvili, V.: Fourier-Haar coefficients of continuous functions. Acta Math. Hungar. 132, 1–14 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
- 11.V. Sh. Tsagareishvili, On the Fourier coefficients of functions with respect to general orthonormal systems, Izv. Ross. Akad. Nauk Ser. Mat., 81 (2017), 183–202 (in Russian); translation in Izv. Math., 81 (2017), 179–198.Google Scholar
- 12.P. L. Ul'janov, On Haar series, Mat. Sb. (N.S.), 63 (1964), 356–391 (in Russian)Google Scholar
- 13.A. Zygmund, Trigonometric Series, I, Cambrige University Press (Cambridge, 1959)Google Scholar
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