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

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.

Authors:

• Snezhana G. Gocheva-Ilieva
• Ivan H. Feschiev

Keywords:

• Effective summation of series
• Favard constants
• finite recursive representations
• Fourier series

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.