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L-Splines

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Abstract

In this paper, we study the problem of unique interpolation and approximation by a class of spline functions,L-splines, containing as special cases the deficient and generalized spline functions ofAhlberg, Nilson, andWalsh [3, 5, 6], the Chebyshevian spline functions ofKarlin andZiegler [27], and the piecewise Hermite polynomial functions, as considered in [17]. We first give sufficient conditions for unique interpolation byL-spline functions in Section 2. Then, we obtain newL ∞ andL 2 error estimates for interpolation byL-splines in Section 4, and show that these error estimates are, in a certain sense, sharp. In addition, we make a similar study for theg-splines ofSchoenberg, cf. [44, 3], in Section 5. In Section 6, an application of these new error estimates is made to the analysis of the error made in the use of finite dimensional subspaces ofL-splines andg-splines. in the Rayleigh-Ritz procedure for the class of nonlinear two-point boundary value problems studied in [17].

Because of the rapid growth of the number of papers devoted to or connected with the topic of splines, we believe that a compilation of papers on splines for the reader's use is desirable, and such a list is found in the References at the end of this paper.

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Authors and Affiliations

  1. Case Western Reserve University, 10900 Euclid Avenue, 44106, Cleveland, Ohio, USA

    M. H. Schultz & R. S. Varga

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  1. M. H. Schultz
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  2. R. S. Varga
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Additional information

This research was supported in part by NSF Grant GP-5553

Papers not specifically concerned with splines are referred to in the text by [1′, 2′], etc.

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Schultz, M.H., Varga, R.S. L-Splines. Numer. Math. 10, 345–369 (1967). https://doi.org/10.1007/BF02162033

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