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Orthogonal Polynomial Projection Error Measured in Sobolev Norms in the Unit Disk

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Abstract

We study approximation properties of weighted \(L^2\)-orthogonal projectors onto the space of polynomials of degree less than or equal to N on the unit disk where the weight is of the generalized Gegenbauer form \(x \mapsto (1-\left|x\right|^2)^\alpha \). The approximation properties are measured in Sobolev-type norms involving canonical weak derivatives, all measured in the same weighted \(L^2\) norm. Our basic tool consists in the analysis of orthogonal expansions with respect to Zernike polynomials. The sharpness of the main result is proved in some cases. A number of auxiliary results of independent interest are obtained including some properties of the uniformly and nonuniformly weighted Sobolev spaces involved, connection coefficients between Zernike polynomials, an inverse inequality, and relations between the Fourier–Zernike expansions of a function and its derivatives.

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Acknowledgments

We thank the anonymous referees for their helpful and constructive suggestions.

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Authors and Affiliations

  1. CI²MA and Departamento de Ingeniería Matemática, Universidad de Concepción, Casilla 160-C, Concepción, Chile

    Leonardo E. Figueroa

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  1. Leonardo E. Figueroa
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Correspondence to Leonardo E. Figueroa.

Additional information

Communicated by Yuan Xu.

The author was supported by the MECESUP project UCO-0713 and the CONICYT project FONDECYT Regular 1130923.

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Figueroa, L.E. Orthogonal Polynomial Projection Error Measured in Sobolev Norms in the Unit Disk. Constr Approx 46, 171–197 (2017). https://doi.org/10.1007/s00365-016-9358-y

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  • DOI: https://doi.org/10.1007/s00365-016-9358-y

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