Abstract
In this paper, we are concerned with multivariate Gegenbauer approximation of functions defined in the d-dimensional hypercube. Two new and sharper bounds for the coefficients of multivariate Gegenbauer expansion of analytic functions are presented based on two different extensions of the Bernstein ellipse. We then establish an explicit error bound for the multivariate Gegenbauer approximation associated with an ℓq ball index set in the uniform norm. We also consider the multivariate approximation of functions with finite regularity and derive the associated error bound on the full grid in the uniform norm. As an application, we extend our arguments to obtain some new tight bounds for the coefficients of tensorized Legendre expansions in the context of polynomial approximation of parametrized PDEs.
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Acknowledgments
We thank two anonymous referees for their careful reading and constructive suggestions. Haiyong Wang also thanks the hospitality of the School of Mathematical Sciences at Fudan University where the present research was initiated.
Funding
Haiyong Wang was supported by National Natural Science Foundation of China under grant number 11671160. Lun Zhang was partially supported by National Natural Science Foundation of China under grant number 11822104, by The Program for Professor of Special Appointment (Eastern Scholar) at Shanghai Institutions of Higher Learning, and by Grant EZH1411513 from Fudan University.
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Communicated by: Yuesheng Xu
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Wang, H., Zhang, L. Analysis of multivariate Gegenbauer approximation in the hypercube. Adv Comput Math 46, 53 (2020). https://doi.org/10.1007/s10444-020-09792-0
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DOI: https://doi.org/10.1007/s10444-020-09792-0