Abstract
The stability of the representation of finite rank operators in terms of a basis is analyzed. A conditioning is introduced as a measure of the stability properties. This conditioning improves some other conditionings because it is closer to the Lebesgue function. Improved bounds for the conditioning of the Fourier sums with respect to an orthogonal basis are obtained, in particular, for Legendre, Chebyshev, and disk polynomials. The Lagrange and Newton formulae for the interpolating polynomial are also considered.
Article PDF
Similar content being viewed by others
References
Berrut, J.P., Trefethen, L.N.: Barycentric Lagrange interpolation. SIAM Rev. 46, 501–517 (2004)
de Boor, C.: A practical guide to splines, Revised Springer Verlag, New York (2001)
Carnicer, J.M., Khiar, Y., Peña, J.M.: Optimal stability of the Lagrange formula and conditioning of the Newton formula. J. Approx. Theory 238, 52–66 (2019)
Carnicer, J.M., Khiar, Y., Peña, J.M.: Central orderings for the Newton interpolation formula. BIT 59(2), 371–386 (2019)
Carnicer, J. M.; Khiar, Y.; Peña, J. M. Conditioning of polynomial Fourier sums. Calcolo 56, no. 3, Paper No. 24 (2019)
Carnicer, J.M., Mainar, E., Peña, J.M.: Stability properties of disk polynomials. Numer. Algorithms 87(1), 119–135 (2021)
Dunkl, C.F., Xu, Y.: Orthogonal polynomials of several variables, Second Edition, Encyclopedia of Mathematics and its applications, 155. Cambridge University Press, Cambridge (2014)
Lam, D. H.; Cuong, L. N.; Van Manh, P.; Van Minh, N., On the conditioning of the Newton formula for Lagrange interpolation. J. Math. Anal. Appl. 505, no. 1, Paper No. 125473, 14 pp (2022)
Rudin, W.: Real and complex analysis. Mac Graw-Hill, London (1970)
Szegő, G.: Orthogonal polynomials, Colloquium Publ, vol. 23. American Mathematical Society, Providence, Rhode Island (2003)
Funding
Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. This work was partially supported by Spanish research grants PGC2018-096321-B-I00, RED2022-134176-T (MCI/AEI) and by Gobierno de Aragón (E41\(\_\)23R).
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of interest
The authors declare no competing interests.
Additional information
Communicated by: Robert Schaback
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Carnicer, J.M., Mainar, E. & Peña, J.M. On the stability of the representation of finite rank operators. Adv Comput Math 49, 52 (2023). https://doi.org/10.1007/s10444-023-10057-9
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10444-023-10057-9