Unicity of best mean approximation by second order splines with variable knots

Authors:
D. L. Barrow, C. K. Chui, P. W. Smith and J. D. Ward

Journal:
Math. Comp. **32** (1978), 1131-1143

MSC:
Primary 41A15

DOI:
https://doi.org/10.1090/S0025-5718-1978-0481754-1

MathSciNet review:
0481754

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Abstract: Let denote the nonlinear manifold of second order splines defined on [0, 1] having at most interior knots, counting multiplicities. We consider the ques tion of unicity of best approximations to a function by elements of . Approximation relative to the norm is treated first, with the results then extended to the best and best one-sided approximation problems. The conclusions in each case are essentially the same, and can be summarized as follows: a sufficiently smooth function satisfying has a unique best approximant from provided either is concave, or is sufficiently large, ; for any , there is a smooth function , with , having at least two best approximants. A principal tool in the analysis is the finite dimensional topological degree of a mapping.

**[1]**D. L. Barrow, C. K. Chui, P. W. Smith, and J. D. Ward,*Unicity of best 𝐿₂ approximation by second-order splines with variable knots*, Bull. Amer. Math. Soc.**83**(1977), no. 5, 1049–1050. MR**0447889**, https://doi.org/10.1090/S0002-9904-1977-14377-2**[2]**Carl de Boor,*Splines as linear combinations of 𝐵-splines. A survey*, Approximation theory, II (Proc. Internat. Sympos., Univ.#Texas, Austin, Tex., 1976) Academic Press, New York, 1976, pp. 1–47. MR**0467092****[3]**C. DE BOOR, "On the approximation by polynomials," in*Approximation with Special Emphasis on Spline Functions*(I. J. Schoenberg, Ed.), Academic Press, New York, 1969, pp. 157-183.**[4]**Dietrich Braess,*On the nonuniqueness of monosplines with least 𝐿₂-norm*, J. Approximation Theory**12**(1974), 91–93. MR**0358149****[5]**Charles K. Chui, Philip W. Smith, and Joseph D. Ward,*On the smoothness of best 𝐿₂ approximants from nonlinear spline manifolds*, Math. Comp.**31**(1977), no. 137, 17–23. MR**0422955**, https://doi.org/10.1090/S0025-5718-1977-0422955-7**[6]**Samuel Karlin, Charles A. Micchelli, Allan Pinkus, and I. J. Schoenberg (eds.),*Studies in spline functions and approximation theory*, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1976. MR**0393934****[7]**Günter Meinardus,*Approximation of functions: Theory and numerical methods*, Expanded translation of the German edition. Translated by Larry L. Schumaker. Springer Tracts in Natural Philosophy, Vol. 13, Springer-Verlag New York, Inc., New York, 1967. MR**0217482****[8]**J. T. Schwartz,*Nonlinear functional analysis*, Gordon and Breach Science Publishers, New York-London-Paris, 1969. Notes by H. Fattorini, R. Nirenberg and H. Porta, with an additional chapter by Hermann Karcher; Notes on Mathematics and its Applications. MR**0433481****[9]**Philip W. Smith,*On the smoothness of local best 𝐿_{𝑝} spline approximations*, Approximation theory, II (Proc. Internat. Sympos., Univ. Texas, Austin, Tex., 1976) Academic Press, New York, 1976, pp. 563–566. MR**0435675**

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DOI:
https://doi.org/10.1090/S0025-5718-1978-0481754-1

Article copyright:
© Copyright 1978
American Mathematical Society