Summability of Fourier orthogonal series for Jacobi weight functions on the simplex in $\mathbb {R}^d$
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- Proc. Amer. Math. Soc. 126 (1998), 3027-3036 Request permission
Abstract:
We study the Fourier expansion of a function in orthogonal polynomial series with respect to the weight functions \[ x_{1}^{\alpha _{1} -1/2} \cdots x_{d}^{\alpha _{d} -1/2}(1-|\mathbf {x}|_{1})^{\alpha _{d+1}-1/2}\] 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
- Richard Askey, Orthogonal polynomials and special functions, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1975. MR 0481145, DOI 10.1137/1.9781611970470
- Hubert Berens and Yuan Xu, Fejér means for multivariate Fourier series, Math. Z. 221 (1996), no. 3, 449–465. MR 1381592, DOI 10.1007/PL00004254
- Charles F. Dunkl, Orthogonal polynomials with symmetry of order three, Canad. J. Math. 36 (1984), no. 4, 685–717. MR 756539, DOI 10.4153/CJM-1984-040-1
- Charles F. Dunkl, Differential-difference operators associated to reflection groups, Trans. Amer. Math. Soc. 311 (1989), no. 1, 167–183. MR 951883, DOI 10.1090/S0002-9947-1989-0951883-8
- Charles F. Dunkl, Integral kernels with reflection group invariance, Canad. J. Math. 43 (1991), no. 6, 1213–1227. MR 1145585, DOI 10.4153/CJM-1991-069-8
- George Gasper, Positive sums of the classical orthogonal polynomials, SIAM J. Math. Anal. 8 (1977), no. 3, 423–447. MR 432946, DOI 10.1137/0508032
- A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and series. Vol. 1, Gordon & Breach Science Publishers, New York, 1986. Elementary functions; Translated from the Russian and with a preface by N. M. Queen. MR 874986
- Gábor Szegő, Orthogonal polynomials, 4th ed., American Mathematical Society Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, R.I., 1975. MR 0372517
- 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.
- Y. Xu, Orthogonal polynomials for a family of product weight functions on the spheres, Canadian J. Math. 49 (1997), 175-192.
- Yuan Xu, Integration of the intertwining operator for $h$-harmonic polynomials associated to reflection groups, Proc. Amer. Math. Soc. 125 (1997), no. 10, 2963–2973. MR 1402890, DOI 10.1090/S0002-9939-97-03986-5
- Y. Xu, Orthogonal polynomials and cubature formulae on spheres and on simplices (to appear).
- A. Zygmund, Trigonometric series: Vols. I, II, Cambridge University Press, London-New York, 1968. Second edition, reprinted with corrections and some additions. MR 0236587
Additional Information
- Yuan Xu
- Affiliation: Department of Mathematics, University of Oregon, Eugene, Oregon 97403-1222
- MR Author ID: 227532
- Email: yuan@math.uoregon.edu
- 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
- © Copyright 1998 American Mathematical Society
- 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