Summability of Fourier orthogonal series for Jacobi weight functions on the simplex in

Author:
Yuan Xu

Journal:
Proc. Amer. Math. Soc. **126** (1998), 3027-3036

MSC (1991):
Primary 33C50, 42C05, 41A63

DOI:
https://doi.org/10.1090/S0002-9939-98-04415-3

MathSciNet review:
1452834

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We study the Fourier expansion of a function in orthogonal polynomial series with respect to the weight functions

on the standard simplex in . It is proved that such an expansion is uniformly summable on the simplex for any continuous function if and only if . Moreover, it is shown that means define a positive linear polynomial identity, and the index is sharp in the sense that means are not positive for .

**1.**R. Askey,*Orthogonal polynomials and special functions*, SIAM, Philadelphia, 1975. MR**58:1288****2.**H. Berens and Y. Xu,*Fejér means for multivariate Fourier series*, Math. Z.**221**(1996), 449-465. MR**97a:42003****3.**C. Dunkl,*Orthogonal polynomials with symmetry of order three*, Can. J. Math.**36**(1984), 685-717. MR**86h:33003****4.**C. Dunkl,*Differential-difference operators associated to reflection groups*, Trans. Amer. Math. Soc.**311**(1989), 167-183; errata, Math. Comp.**66**(1997), 1765-1766. MR**90k:33027****5.**C. Dunkl,*Integral kernels with reflection group invariance*, Can. J. Math.**43**(1991), 1213-1227. MR**93g:33012****6.**G. Gasper,*Positive sums of the classical orthogonal polynomials*, SIAM J. Math. Anal.**8**(1977), 423-447. MR**55:5925****7.**A. P. Prudnikov, Yu. A. Brychkov and O. I. Marichev,*Integrals and Series, Vol. 1: Elementary Functions*, Gordon and Breach Sci. Publ., New York, 1986. MR**88f:00013**; CMP**97:08****8.**G. Szeg\H{o},*Orthogonal polynomials*, 4th ed., Amer. Math. Soc. Colloq. Publ. vol.23, Providence, RI, 1975. MR**51:8724****9.**Y. Xu,*On orthogonal polynomials in several variables*, Special functions, -series, and related topics, Fields Institute Communications Series, vol. 14, 1997, pp. 247-270. CMP**97:12****10.**Y. Xu,*Orthogonal polynomials for a family of product weight functions on the spheres*, Canadian J. Math.**49**(1997), 175-192. CMP**97:09****11.**Y. Xu,*Integration of the intertwining operator for -harmonic polynomials associated to reflection groups*, Proc. Amer. Math. Soc.**125**(1997), 2963-2973. MR**97m:33004****12.**Y. Xu,*Orthogonal polynomials and cubature formulae on spheres and on simplices*(to appear).**13.**A. Zygmund,*Trigonometric Series*, 2nd ed., Cambridge Univ. Press, Cambridge, 1968. MR**38:4882**

Retrieve articles in *Proceedings of the American Mathematical Society*
with MSC (1991):
33C50,
42C05,
41A63

Retrieve articles in all journals with MSC (1991): 33C50, 42C05, 41A63

Additional Information

**Yuan Xu**

Affiliation:
Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222

Email:
yuan@math.uoregon.edu

DOI:
https://doi.org/10.1090/S0002-9939-98-04415-3

Keywords:
Orthogonal polynomials in several variables,
on simplex,
Ces\`{a}ro summability,
positive kernel

Received by editor(s):
March 14, 1997

Additional Notes:
Supported by the National Science Foundation under Grant DMS-9500532.

Communicated by:
Palle E. T. Jorgensen

Article copyright:
© Copyright 1998
American Mathematical Society