Abstract
We investigate the convergence of the linear means of the Fourier-Jacobi series of functions ƒ(x) from the weight space L α,β for x = 1 for the case in which this point is a Lebesgue point for ƒ. We establish su.cient summability conditions depending on the behavior of the function on the closed interval [−1, 0] and on the properties of the matrix involved in the summation method.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Bibliography
G. Szegö, Orthogonal Polynomials, Colloquium Publ., vol. XXIII, Amer. Math. Soc., Providence, RI, 1959; Russian transl.: Moscow, Fizmatgiz, 1962.
S. B. Topuriya, Fourier-Laplace Series on the Sphere [in Russian], Tbilisi Univ., Tbilisi, 1987.
S. G. Kal’nei, “Summability of Jacobi series by triangular matrices,” Mat. Zametki [Math. Notes], 34 (1983), no. 1, 91–103.
E. Kogbetliantz, “Über die (C, δ) Summierbarkeit der Laplaceschen Reihe fur 1− < δ < 1,” Math. Z., 14 (1922), 99–109.
S. G. Kal’nei, “On the summability of Jacobi series at Lebesgue points,” Analysis Mathematica, 29 (2003), 181–194.
S. G. Kal’nei, “Approximation of the linear means of Jacobi series at Lebesgue points,” in: Approximation of Functions: Theoretical and Applied Aspects, Collection of Papers Devoted to the Memory of Professor A. V. E.mov, MIÉT, Moscow, 2003, pp. 124–139.
S. G. Kal’nei, “On linear methods for the summation of Jacobi series for half-integer α, ” Analysis Mathematica, 22 (1996), 35–50.
S. G. Kal’nei, “Lower bound of the Lebesgue function of linear means of Fourier-Jacobi series,” Trudy Mat. Inst. Steklov [Proc. Steklov Inst. Math.], 170 (1984), 113–118.
S. G. Kal’nei, “On an analog of S. M. Nikolskii’s theorem for Jacobi series,” Ukrain. Mat. Zh. [Ukrainian Math. J.], 43 (1991), no. 4, 503–513.
A. Bonami and J.-L. Clerc, “Sommes de Cesàro et multiplicateurs des développements en harmoniques sphériques,” Trans. Amer. Math. Soc., 183 (1973), 223–263.
I. P. Natanson, The Theory of Functions of a Real Variable [in Russian], Nauka, Moscow, 1974.
Author information
Authors and Affiliations
Additional information
__________
Translated from Matematicheskie Zametki, vol. 80, no. 2, 2006, pp. 193–203.
Original Russian Text Copyright © 2006 by S. G. Kal’nei.
Rights and permissions
About this article
Cite this article
Kal’nei, S.G. On the convergence of the linear means of Jacobi series at Lebesgue points in the case of half-integer α. Math Notes 80, 188–198 (2006). https://doi.org/10.1007/s11006-006-0127-2
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s11006-006-0127-2