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

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 ↦ ( 1 - x 2 ) α . The approximation properties are measured in Sobolev-type norms involvi...

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Bibliographic Details
Published inConstructive approximation Vol. 46; no. 1; pp. 171 - 197
Main Author Figueroa, Leonardo E.
Format Journal Article
LanguageEnglish
Published New York Springer US 01.08.2017
Springer Nature B.V
Subjects
Analysis
Approximation
Derivatives
Error analysis
Fourier analysis
Mathematical analysis
Mathematics
Mathematics and Statistics
Norms
Numerical Analysis
Polynomials
Projection
Projectors
Properties (attributes)
Sharpness
Zernike polynomials
Orthogonal projection
42C10
Zernike polynomials
Weighted Sobolev space
Connection coefficients
41A25
46E35
41A10
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Summary: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 ↦ ( 1 - x 2 ) α . 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.
ISSN:0176-4276
1432-0940
DOI:10.1007/s00365-016-9358-y