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Journal of Fourier Analysis and Applications

, Volume 24, Issue 6, pp 1438–1459 | Cite as

Approximation by Polynomials in Sobolev Spaces with Jacobi Weight

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Abstract

Polynomial approximation is studied in the Sobolev space \(W_p^r(w_{\alpha ,\beta })\) that consists of functions whose r-th derivatives are in weighted \(L^p\) space with the Jacobi weight function \(w_{\alpha ,\beta }\). This requires simultaneous approximation of a function and its consecutive derivatives up to s-th order with \(s \le r\). We provide sharp error estimates given in terms of \(E_n(f^{(r)})_{L^p(w_{\alpha ,\beta })}\), the error of best approximation to \(f^{(r)}\) by polynomials in \(L^p(w_{\alpha ,\beta })\), and an explicit construction of the polynomials that approximate simultaneously with the sharp error estimates.

Keywords

Approximation Simultaneous approximation Sobolev space Jacobi weight 

Mathematics Subject Classification

41A10 41A25 42C05 42C10 33C45 

Notes

Acknowledgements

The author thanks Danny Leviatan for his careful readings and corrections. The author was supported in part by NSF Grant DMS-1510296.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of OregonEugeneUSA

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