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Estimates of the maximal Cesaro operators of the weighted orthogonal polynomial expansions in several variables Open Access

Descriptions

Other title
Subject/Keyword
orthogonal polynomial expansions
almost everywhere convergence
maximal Cesaro means
spherical h-harmonics
ball
sphere
simplex
Type of item
Thesis
Degree grantor
University of Alberta
Author or creator
Ye, Wenrui
Supervisor and department
Dai, Feng
Examining committee member and department
Safouhi, Hassan (Mathematical and Statistical Sciences)
Lau, Tony (Mathematical and Statistical Sciences)
Dai, Feng (Mathematical and Statistical Sciences)
Han, Bin (Mathematical and Statistical Sciences)
Department
Department of Mathematical and Statistical Sciences
Specialization
Mathematics
Date accepted
2013-09-24T14:37:40Z
Graduation date
2013-11
Degree
Master of Science
Degree level
Master's
Abstract
Estimates of the maximal Cesaro means at the "critical index'' are established for the orthogonal polynomial expansions (OPEs) with respect to the weight function on the unit sphere. These estimates allow us to improve several known results in this area, including the almost everywhere (a.e.) convergence of the Cesaro means at the "critical index'', the sufficient conditions for the Marcinkiewitcz multiplier theorem, and a Fefferman-Stein type inequality for the Cesaro operators. In addition, several similar results for the weighted OPEs on the unit ball and on the simplex are deduced from the corresponding weighted results on the sphere.
Language
English
DOI
doi:10.7939/R3MC8RS8J
Rights
This thesis is made available by the University of Alberta Libraries with permission of the copyright owner solely for the purpose of private, scholarly or scientific research. This thesis, or any portion thereof, may not otherwise be copied or reproduced without the written consent of the copyright owner, except to the extent permitted by Canadian copyright law.
Citation for previous publication
F. Dai, S. Wang and W. Ye, Estimates of the maximal Cesaro operators of the weighted orthogonal polynomial expansions in several variables, J. Funct. Anal., 265 (2013), 2357 - 2387.R. Askey, G. Andrews and R. Roy, Special Functions, Encyclopedia of Mathematics and its Applications 71, Cambridge University Press, Cambridge, 1999.A. Benedek, R. Panzone, The space L^{p} with mixed norm. Duke Math. J. 28(1961), 301--324.A. Bonami and J.L. Clerc, Sommes de Cesaro et multiplicateurs des developpments en harmonque sphrique, Trans. Amer. Math. Soc., 183 (1973), 223-263.F. Dai, Multivariate polynomial inequalities with respect to doubling weights and A_{\infty} weights, J. Funct. Anal. 235 (2006), no. 1,137--170.F. Dai and Y. Xu, Maximal function and multiplier theorem for weighted space on the unit sphere, J. Funct. Anal., 249 (2007), 477 - 504.F. Dai, Y. Xu, Cesaro Means of Orthogonal Expansions in Several Variables, Constr. Appprox., 29 (2009): 129-155.C.F. Dunkl, Differential-difference Operators Associated to Reflection Groups, Trans. Amer. Math. Soc. 311 (1989): 167-183.C.F. Dunkl, Integral Kernels with Reflection Group Invariance, Can.J.Math. 43 (1991):1213-1227.C.F. Dunkl, Y.Xu, Orthogonal Polynomials of Several Variables, Cambridge Univ. Press, Cambridge (2001).A. Erdedlyi, W. Magnus, F. Oberhettinger, F.G. Tricomi, Higher transcendental functions, McGraw-Hill, Vol 2, New York, 1953.C. Fefferman and E. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107-115.L. Grafakos, Classical and Modern Fourier Analysis, Pearson Education Inc., Upper Saddle River, New Jersey, 2004.Z.K. Li, Y. Xu, Summability of orthogonal expansions of several variables, J. Approx. Theory, 122 (2003): 267-333.C. Markett, Cohen type inequalities for Jacobi, Laguerre and Hermite expansions, SIAM J. Math. Anal. 14(1983): 819-833.C. Meaney, Divergent Sums of Spherical Harmonics, in: International Conference on Harmonic Analysis and Related Topics, Macquarie University, January 2002, Proc. Centre Math. Appl. Austral. Univ. 41(2003): 110-117.S. Pawelke, Uber Approximationsordnung bei Kugelfunktionen und algebraischen Polynomen, Tohoku Math.I., 24: 473-486.E.M. Stein, On limits of seqences of operators, Ann. of Math. (2) 74, 1961, 140--170.E.M. Stein, Interpolation of linear operators, Trans. Amer. Math. Soc. 83 (1956), 482--492.E. M. Stein, Topics in Harmonic Analysis Related to the Littlewood-Paley Theory, Princeton Univ. Press, Princeton, NJ, 1970.M.H. Taibleson, Estimates for finite expansions of Gegenbauer and Jacobi polynomials. Recent progress in Fourier analysis (El Escorial, 1983), 245 - 253, North-Holland Math. Stud., 111, North-Holland, Amsterdam, 1985.K. Wang and L. Li, Harmonic Analysis and Approximation on the Unit Sphere, Science Press, Beijing, 2000.G. Szego, Orthogonal Polynomials, Amer. Math. Soc. Colloq. Publ. Vol.23, Providence, 4th edition, 1975.Y. Xu, Integration of the Intertwining Operator for h-harmonic Polynomials Associated to Reflection Groups, Proc. Amer. Math. Soc., 125 (1997): 2963-2973.Y. Xu, Almost Everywhere Convergence of Orthogonal Expansions of Several Variables, Constr. Approx., 22 (2005): 67-93.Y. Xu, Weighted approximation of functions on the unit sphere, Constr. Approx., 21 (2005), 1-28.Y. Xu, Approximation by means of h-harmonic polynomials on the uint sphere, Adv. in Comput. Math., 2004, 21: 37-58.Y. Xu, Orthogonal polynomials and summability in Fourier orthogonal series on spheres and on balls, Math. Proc. Cambridge Philos. Soc., 2001, 31: 139-155.Y. Xu, Orthogonal polynomials for a family of product weight functions on the spheres, Canadian J. Math. 49 (1997), 175-192.Y. Xu, Summability of Fourier orthogonal series for Jacobi weight on a ball in R^d, Trans. Amer. Math. Soc. 351 (1999), 2439-2458.Y. Xu, Summability of Fourier orthogonal series for Jacobi weight function on the simplex in R^d, Proc. Amer. Math. Soc.126 (1998), 3027-3036.H. Zhou, Divergence of Ces\`aro means of spherical h-harmonic expansions, J. Approx. Theory 147 (2007), no. 2, 215--220.A. Zygmund, Trigonometric series: Vols. I, II, Cambridge University Press, London, 1968.

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