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Estimates for Fourier sums and eigenvalues of integral operators via multipliers on the sphere


Authors: T. Jordão and V. A. Menegatto
Journal: Proc. Amer. Math. Soc. 144 (2016), 269-283
MSC (2010): Primary 33C65, 41A17, 41A36, 41A60, 42A16; Secondary 45M05, 45C05, 42B10, 42A82
DOI: https://doi.org/10.1090/proc12716
Published electronically: July 8, 2015
MathSciNet review: 3415595
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Abstract: We provide estimates for weighted Fourier sums of integrable functions defined on the sphere when the weights originate from a multiplier operator acting on the space where the function belongs. That implies refined estimates for weighted Fourier sums of integrable kernels on the sphere that satisfy an abstract Hölder condition based on a parameterized family of multiplier operators defining an approximate identity. This general estimation approach includes an important class of multiplier operators, namely, that defined by convolutions with zonal measures. The estimates are used to obtain decay rates for the eigenvalues of positive integral operators on $ L^2(S^m)$ and generated by a kernel satisfying the Hölder condition based on multiplier operators on $ L^2(S^m)$.


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Additional Information

T. Jordão
Affiliation: Departamento de Matemática, ICMC-USP - São Carlos, Caixa Postal 668, 13560-970 São Carlos SP, Brazil
Email: thsjordao@gmail.com

V. A. Menegatto
Affiliation: Departamento de Matemática, ICMC-USP - São Carlos, Caixa Postal 668, 13560-970 São Carlos SP, Brazil
Email: menegatt@icmc.usp.br

DOI: https://doi.org/10.1090/proc12716
Received by editor(s): May 29, 2014
Received by editor(s) in revised form: December 18, 2014
Published electronically: July 8, 2015
Additional Notes: The authors were partially supported by FAPESP, grants $#$ 2011/21300-7 and $#$ 2014/06209-1
Communicated by: Walter Van Assche
Article copyright: © Copyright 2015 American Mathematical Society