New Recursive Representations for the Favard Constants with Application to the Summation of Series

Authors: Snezhana G. Gocheva-Ilieva, Ivan H. Feschiev

Abstract:

In this study integral form and new recursive formulas for Favard constants and some connected with them numeric and Fourier series are obtained. The method is based on preliminary integration of Fourier series which allows for establishing finite recursive representations for the summation. It is shown that the derived recursive representations are numerically more effective than known representations of the considered objects.

Keywords: Effective summation of series, Favard constants, finite recursive representations, Fourier series

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1085393

References:

[1] N. P., Korneichuk, Exact constants in approximation theory, New York: Cambrige Univ. Pres, 1991, ch. 3, 4.
[2] A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and series: Elementary functions, Boca Raton: CRC Press, 1998, ch. 5.
[3] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, with Formulas, Graphs, and Mathematical Tables, New York: Dover, 9th ed., 1964, ch. 23.
[4] J. Favard, "Sur les meilleurs precedes d'approximation de certaines classes de fonctions par des polynomes trigonometriques," Bull. Sci. Math., vol. 61, pp. 209-224 and 243-256, 1937.
[5] S. R. Finch, Mathematical constants, New York: Cambridge Univ. Press, pp. 255-257, 2003.
[6] J. Bustamante, Algebraic Approximation: A Guide to Past and Current Solutions, Basel: Springer Basel AG, 2012, pp. 4-5, 14-18, 101-111.
[7] S. Foucart, Y. Kryakin, and A. Shadrin, "On the exact constant in the Jackson-Stechkin inequality for the uniform metric," Constr. Approx., vol. 29, pp. 157-179, 2009.
[8] Yu. N. Subbotin and S. A. Telyakovskii, "On the equality of Kolmogorov and relative widths of classes of differentiable functions", Math. Notes, vol. 86, pp. 432-439, 2009.
[9] R. A. DeVore, G. G. Lorentz, Constructive Approximation, Berlin: Springer-Verlag, 1993, pp. 148-157, 212-215.
[10] V. F. Babenko and V. A. Zontov, "Bernstein-type inequalities for splines defined on the real axis," Ukr. Math. J., vol. 63, pp. 699-708, 2011.
[11] G. Vainikko, "Error estimates for the cardinal spline interpolation", Z. Anal. Anwend., vol. 28, pp. 205-222, 2009.
[12] L. A. Apaicheva, "Optimal quadrature and cubature formulas for singular integrals with Hilbert kernels," Russian Math. (Iz. VUZ), vol. 48, pp. 14-25, 2004.
[13] F. D. Gakhov and I. Kh. Feschiev, "Approximate calculation of singular integrals," Izv. Akad. Nauk BSSR, Ser. Fiz. Mat. Nauk, vol. 4, pp. 5-12, 1977.
[14] F. D. Gakhov and I. Kh. Feschiev, "Interpolation of Singular Integrals and an Approximate Solution of the Riemann Problem," Vestsi Akad. Nauk BSSR, Ser. Fiz.-Mat. Nauk, No. 5, pp. 3-13, 1982.
[15] B. G. Gabdulkhaev, "Finite-dimensional approximations of singular integrals and direct methods of solution of singular integral and integrodifferential equations," Journal of Soviet Mathematics, vol. 18, pp. 593- 627, March 1982.
[16] H. Brass and K. Petras, Quadrature Theory: The Theory of Numerical Integration on a Compact Interval, Providence: Amer. Math. Soc., 2011, ch. 4, 5.
[17] A. V. Mironenko, "On the Jackson-Stechkin inequality for algebraic polynomials", Proc. Inst. Math. Mech., vol. 273, suppl. 1, pp. S116- S123, 2011.
[18] E. W. Weisstein, Favard constants, available on-line at: http://mathworld.wolfram.com/FavardConstants.html, accessed 27 Dec 2012.
[19] S. Wolfram, The Mathematica Book, 5th ed., Champaign: Wolfram Media, Inc. 2003.