Abstract
Lubinsky and Totik’s decomposition [11] of the Cesàro operators σ (α,β) n of Jacobi expansions is modified to prove uniform boundedness in weighted sup norms, i.e., ‖w (a,b) σ (α,β) n ‖∞ ≦ C‖w (a,b) f‖∞, whenever α,β ≧ −1/2 and a, b are within the square around (α/2 + 1/4, α/2 + 1/4) having a side length of 1. This approach uses only classical results from the theory of orthogonal polynomials and various estimates for the Jacobi weights. The present paper is concerned with the main theorems and ideas, while a second paper [7] provides some necessary estimations.
Similar content being viewed by others
References
R. Askey and I. I. Hirschman, Jr., Mean summability for ultraspherical polynomials, Math. Scand., 12 (1963), 167–177.
Sagun Chanillo and Benjamin Muckenhoupt, Weak Type Estimates for Cesàro Sums of Jacobi Polynomial series, Mem. Amer. Math. Soc., 102 (1993), no. 487, viii-90. MR 1132609 (93g:42018).
W. Chen and Z. Ditzian, Best approximation and K-functionals, Acta Math. Hungar., 75 (1997), 165–208.
Z. Ditzian, A K-functional and the rate of convergence of some linear polynomial operators, Proc. Amer. Math. Soc., 124 (1996), 1773–1781.
M. Felten, Conjectures on mean convergence of Jacobi Cesàro series, submitted, 1–9.
M. Felten, Multiplier theorems for finite sums of Jacobi polynomials, submitted, 1–9.
M. Felten, Uniform boundedness of (C, 1) means of Jacobi expansions in weighted sup norms. II (Some necessary estimations), Acta Math. Hungar. (to appear).
M. Felten, Boundedness of first order Cèsaro means in Jacobi spaces and weighted approximation on [−1, 1], 2004, Habilitationsschrift, Seminarberichte aus dem Fachbereich Mathematik der FernUniversität in Hagen (ISSN 0944-5838), Band 75, pp. 1–170.
M. Felten, Local estimates for Jacobi polynomials, Journal of Inequalities in Pure and Applied Mathematics, 8 (2007), Article 3, 7 pp.
G. Freud, Orthogonale Polynome, Birkhäuser Verlag (Basel, 1969), Lehrbücher und Monographien aus dem Gebiete der Exakten Wissenschaften, Mathematische Reihe, Band 33.
D. S. Lubinsky and V. Totik, Best weighted polynomial approximation via Jacobi expansions, SIAM J. Math. Anal., 25 (1994), 555–570.
P. G. Nevai, Orthogonal Polynomials, Mem. Amer. Math. Soc. 18 (1979), no. 213, pp. v–185.
J. Szabados, Weighted error estimates for approximation by Cesàro means of Fourier-Jacobi series in spaces of locally continuous functions, Analysis Math. (to appear).
G. Szegő, Orthogonal Polynomials, fourth ed., American Mathematical Society (Providence, R.I., 1975), Colloquium Publications, Vol. XXIII.
L. Szili and P. Vértesi, On summability of weighted Lagrange interpolation. III Jacobi weights, Acta Math. Hungar., 104 (2004), 39–62. MR 2069961 (2005i:41004).
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Felten, M. Uniform boundedness of (C, 1) means of Jacobi expansions in weighted sup norms. I (The main theorems and ideas). Acta Math Hung 118, 227–263 (2008). https://doi.org/10.1007/s10474-007-6207-2
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10474-007-6207-2
Key words and phrases
- Cesàro means
- Jacobi expansions
- uniform boundedness
- weighted polynomial approximation
- orthogonal polynomials