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Constructive Approximation

, Volume 46, Issue 1, pp 171–197 | Cite as

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

  • Leonardo E. Figueroa
Article

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.

Keywords

Zernike polynomials Connection coefficients Orthogonal projection Weighted Sobolev space 

Mathematics Subject Classification

41A25 41A10 42C10 46E35 

Notes

Acknowledgments

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

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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.CI²MA and Departamento de Ingeniería MatemáticaUniversidad de ConcepciónConcepciónChile

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