Detection of singularities using segment approximation
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- by R. Grothmann and H. N. Mhaskar PDF
- Math. Comp. 59 (1992), 533-540 Request permission
Abstract:
We discuss best segment approximation (with free knots) by polynomials to piecewise analytic functions on a real interval. It is shown that, if the degree of the polynomials tends to infinity and the number of knots is the same as the number of singularities of the function, then the optimal knots converge geometrically fast to the singularities. When the degree is held fixed and the number of knots tends to infinity, we study the asymptotic distribution of the optimal knots.References
-
S. N. Bernstein, Sur la valeur asymptotique de la meilleure approximation de $|x|$, Acta Math. 37 (1913), 1-57.
- Maurice Hasson, Derivatives of the algebraic polynomials of best approximation, J. Approx. Theory 29 (1980), no. 2, 91–102. MR 595594, DOI 10.1016/0021-9045(80)90108-2
- I. P. Na-t’ang-sung, Han-shu kou-tsao lun. Shang ts’e, Science Press, Peking, 1965 (Chinese). Translated from the Russian by Hsü Chia-fu and Cheng Wei-shing. MR 0201878
- G. Nürnberger, M. Sommer, and H. Strauss, An algorithm for segment approximation, Numer. Math. 48 (1986), no. 4, 463–477. MR 834333, DOI 10.1007/BF01389652
- G. Meinardus, G. Nürnberger, M. Sommer, and H. Strauss, Algorithms for piecewise polynomials and splines with free knots, Math. Comp. 53 (1989), no. 187, 235–247. MR 969492, DOI 10.1090/S0025-5718-1989-0969492-7
- A. F. Timan, Theory of approximation of functions of a real variable, A Pergamon Press Book, The Macmillan Company, New York, 1963. Translated from the Russian by J. Berry; English translation edited and editorial preface by J. Cossar. MR 0192238
Additional Information
- © Copyright 1992 American Mathematical Society
- Journal: Math. Comp. 59 (1992), 533-540
- MSC: Primary 41A15; Secondary 41A10, 65D15
- DOI: https://doi.org/10.1090/S0025-5718-1992-1145662-2
- MathSciNet review: 1145662