Cesàro Means of Weighted Orthogonal Expansions on Regular Domains
Abstract
:1. Introduction
2. Preliminaries
2.1. Dunkl Theory and Spherical h-Harmonic Expansions
2.2. Jacobi Polynomials
2.3. Doubling Weights on the Sphere
2.4. Weighted Christoffel Functions
- (i)
- If , we haveSince , we can use the Lemma 4 and follow the same argument as in the proof of [23] (Theorem 3.1, (3.12)) to get
- (ii)
- If , we use the linearity of , Lemma 4, and the fact that
- (iii)
- If , following the same idea as (ii), and using the Lemma 4, we have:
3. Main Results
4. Proofs of Main Theorems
4.1. Proof of Theorem 1
4.2. Proof of Theorem 2
5. Weighted Orthogonal Polynomial Expansions (WOPEs) on the Ball and the Simplex
5.1. WOPEs in Several Variables
5.2. WOPEs on the Unit Ball
5.3. Results on the Ball
5.4. WOPEs on the Simplex
5.5. Results on the Simplex
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Acknowledgments
Conflicts of Interest
References
- Cesàro, E. Sur la multiplication des séries. Bull. Sci. Math. 1890, 14, 114–120. [Google Scholar]
- Dai, F.; Wang, K.Y. A Survey of Polynomial Approximation on the Sphere. Int. J. Wavelets Multiresolut. Inf. Process. 2019, 7, 749–771. [Google Scholar] [CrossRef]
- Sogge, C.D. Oscillatory integrals and spherical harmonics. Duke Math. J. 1986, 53, 43–65. [Google Scholar] [CrossRef]
- Sogge, C.D. On the convergence of Riesz means on compact manifolds. Ann. Math. 1987, 126, 439–447. [Google Scholar] [CrossRef]
- Dunkl, C.F. Differential-difference operators associated to reflection groups. Trans. Am. Math. Soc. 1989, 311, 167–183. [Google Scholar] [CrossRef]
- Dunkl, C.F. Integral kernels with reflection group invariance. Can. J. Math. 1991, 43, 1213–1227. [Google Scholar] [CrossRef]
- Dunkl, C.F. Reflection groups and orthogonal polynomials on the sphere. Math. Z. 1988, 197, 33–60. [Google Scholar] [CrossRef]
- Rösler, M. Positivity of Dunkl’s intertwining operator. Duke Math. J. 1999, 98, 445–463. [Google Scholar] [CrossRef]
- Dunkl, C.F.; Xu, Y. Orthogonal Polynomials of Several Variables; Encyclopedia of Mathematics and Its Applications; Cambridge University Press: Cambridge, UK, 2014. [Google Scholar]
- Dai, F.; Xu, Y. Cesàro means of orthogonal expansions in several variables. Const. Approx. 2009, 29, 129–155. [Google Scholar] [CrossRef] [Green Version]
- Xu, Y. Intertwining operator associated to symmetric groups and summability on the unit sphere. J. Approx. Theory 2021, 272, 105649. [Google Scholar] [CrossRef]
- Dai, F.; Xu, Y. Boundedness of projection operators and Cesàro means in weighted Lp space on the unit sphere. Trans. Am. Math. Soc. 2009, 361, 3189–3221. [Google Scholar] [CrossRef] [Green Version]
- Dai, F.; Wang, S.; Ye, W.R. Maximal estimates for the Cesàro means of weighted orthogonal polynomial expansions on the unit sphere. J. Funct. Anal. 2013, 265, 2357–2387. [Google Scholar] [CrossRef]
- Dai, F.; Ge, Y. Sharp estimates of the Cesàro kernels for weighted orthogonal polynomial expansions in several variables. J. Funct. Anal. 2021, 280, 108865. [Google Scholar] [CrossRef]
- Li, Z.-K.; Xu, Y. Summability of orthogonal expansions of several variables. J. Approx. Theory 2003, 122, 267–333. [Google Scholar] [CrossRef] [Green Version]
- Rösler, M. Dunkl operators: Theory and applications. In Orthogonal Polynomials and Special Functions, Leuven; Springer: Berlin, Germany, 2003; pp. 93–135. [Google Scholar]
- Berens, H.; Schmid, H.J.; Xu, Y. Bernstein-Durrmeyer polynomials on a simplex. J. Approx. Theory 1992, 68, 247–261. [Google Scholar] [CrossRef] [Green Version]
- Xu, Y. Integration of the intertwining operator for h-harmonic polynomials associated to reflection groups. Proc. Am. Math. Soc. 1997, 125, 2963–2973. [Google Scholar] [CrossRef]
- Dunkl, C.F. Intertwining operator associated to the group S3. Trans. Am. Math. Soc. 1995, 347, 3347–3374. [Google Scholar]
- Szegö, G. Orthogonal Polynomials; American Mathematical Society: Providence, RI, USA, 1975; Volume XXIII. [Google Scholar]
- Dai, F.; Xu, Y. Approximation Theory and Harmonic Analysis on Spheres and Balls; Springer Monographs in Mathematics; Springer: New York, NY, USA, 2013. [Google Scholar]
- Dai, F. Multivariate polynomial inequalities with respect to doubling weights and A∞ weights. J. Funct. Anal. 2006, 235, 137–170. [Google Scholar] [CrossRef] [Green Version]
- Dai, F.; Feng, H. Riesz transforms and fractional integration for orthogonal expansions on spheres, balls and simplices. Adv. Math. 2016, 301, 549–614. [Google Scholar] [CrossRef]
- Brown, G.; Dai, F. Approximation of smooth functions on compact two-point homogeneous spaces. J. Funct. Anal. 2005, 220, 401–423. [Google Scholar] [CrossRef] [Green Version]
- Xu, Y. Orthogonal polynomials and summability in Fourier orthogonal series on spheres and on balls. In Mathematical Proceedings of the Cambridge Philosophical Society; Cambridge University Press: Cambridge, UK, 2001; Volume 31, pp. 139–155. [Google Scholar]
- Xu, Y. Orthogonal polynomials and cubature formulae on balls, simplices, and spheres. J. Comput. Appl. Math. 2001, 127, 349–368. [Google Scholar] [CrossRef] [Green Version]
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Feng, H.; Ge, Y. Cesàro Means of Weighted Orthogonal Expansions on Regular Domains. Mathematics 2022, 10, 2108. https://doi.org/10.3390/math10122108
Feng H, Ge Y. Cesàro Means of Weighted Orthogonal Expansions on Regular Domains. Mathematics. 2022; 10(12):2108. https://doi.org/10.3390/math10122108
Chicago/Turabian StyleFeng, Han, and Yan Ge. 2022. "Cesàro Means of Weighted Orthogonal Expansions on Regular Domains" Mathematics 10, no. 12: 2108. https://doi.org/10.3390/math10122108