Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Summability of Fourier orthogonal series for Jacobi weight functions on the simplex in $\mathbb{R}^{d}$

Author: Yuan Xu
Journal: Proc. Amer. Math. Soc. 126 (1998), 3027-3036
MSC (1991): Primary 33C50, 42C05, 41A63
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

\begin{displaymath}x_{1}^{\alpha _{1} -1/2} \cdots x_{d}^{\alpha _{d} -1/2}(1-|\mathbf{x}|_{1})^{\alpha _{d+1}-1/2}\end{displaymath}

on the standard simplex $\Sigma ^{d}$ in $\mathbb{R}^{d}$. It is proved that such an expansion is uniformly $(C, \delta )$ summable on the simplex for any continuous function if and only if $\delta > |\alpha |_{1} + (d-1)/2$. Moreover, it is shown that $(C, |\alpha |_{1} + (d+1)/2)$ means define a positive linear polynomial identity, and the index is sharp in the sense that $(C,\delta )$ means are not positive for $0 <\delta <|\alpha |_{1} + (d+1)/2$.

References [Enhancements On Off] (What's this?)

  • 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, $q$-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 $h$-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

Similar Articles

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

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

American Mathematical Society