Abstract
For certain weight functions p(t) and q(t), upper bounds are obtained for the difference between partial sums of Fourier series of a function/ with respect to the systems σp and σq of polynomials orthogonal on [−1, 1] (a comparison theorem is incidentally proved for the systems σp and σq). By using these upper bounds, known asymptotic expressions for the Lebesgue function, and an upper bound (forf ε Wr Hω) of the remainder in a Fourier-Chebyshev series, we establish corresponding results for Fourier series with respect to a system σp.
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Literature cited
G. Aleksich, The Convergence of Orthogonal Series [in Russian], Moscow (1963).
G. Szegō, Orthogonal Polynomials, American Mathematical Society, New York (1939).
A. Zygmund, Trigonometrical Series, Vol. 1, University Press, Cambridge (1959).
R. A. Hunt, On the Convergence of Fourier Series. Orthogonal Expansions and their Continuous Analogs. London and Amsterdam, Proceedings of the Conference held at Southern Illinois University, Edwardsville, April 27–29 (1967), pp. 235–255.
A. F. Timan, The Approximation of Functions of a Real Variable [in Russian], Moscow (1960).
T. Frey, “Sur un théorème de Korous,” Publ. Math.,7, Nos. 1–4, 320–352 (1960).
G. I. Barkov, “Comparison of the rate growth of orthonormal systems of polynomials,” Trudy Chelyabinskogo Ped. In-ta (Matematika),2, 5–8 (1964).
A. A. Tsygankov, “Comparison of the rate of growth of systems of orthogonal polynomials,” Matem. Zapiski (Ural'skii Un-t), Vol. 6, Tetrad 2, Sverdlovsk, 148–165 (1968).
V. M. Badkov, “Approximation of functions by partial sums of Fourier series with respect to generalized Jacobi polynomials,” Matem. Zametki,3, No. 6, 671–682 (1968).
V. P. Konoplev, “Polynomials orthogonal with a weight function taking the values zero of infinity at distinct points of the orthogonality interval,” Dokl. Akad. Nauk SSSR,141, No. 4, 781–784 (1961).
P. K. Suetin, “Some properties of polynomials orthogonal on an interval,” Sibirsk. Matem. Zh.,10, No. 3, 653–670 (1969).
Ya. L. Geronimus, Polynomials Orthogonal on a Circle and on an Interval [in Russian], Moscow (1958).
A. N. Kolmogorov, “Zur Grossenordnung des Restgliedes Fourierscher Reihen differzierbarer Funktionen,” Ann. of Math.,36, 521–526 (1935).
S. A. Agakhanov and G. I. Natanson, Letter to the Editor, Izv. Vyssh. Uchebn. Zaved., Matem., No. 1 (80), 109–110 (1969).
V. M. Badkov, “Equiconvergence of Fourier series with respect to orthogonal polynomials,” Matem. Zametki,5, No. 3, 285–295 (1969).
H. Pollard, “The mean convergence of orthogonal series,” III, Duke Math. J.,16, 189–191 (1949).
R. Askey and I. I. Hirschman, Jr., “Mean summability for ultraspherical polynomials,” Math. Scand.,12, 167–177 (1963).
B. Muchenhoupt, “Mean convergence of Jacobi series,” Proc. Amer. Math. Soc,23, No. 2, 306–310 (1969).
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Translated from Matematicheskie Zametki, Vol. 8, No. 4, pp. 431–441, October, 1970.
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Badkov, V.M. Approximation of functions by partial sums of Fourier series in polynomials orthogonal on an interval. Mathematical Notes of the Academy of Sciences of the USSR 8, 712–717 (1970). https://doi.org/10.1007/BF01104370
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DOI: https://doi.org/10.1007/BF01104370