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On constrained multivariate splines and their approximations

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Summary

The variational formulation of multivariate spline functions is generalized to include cases where the function has to satisfy inequality constraints such as positivity and convexity. Condition for existence and uniqueness of a solution is given. Approximation to the solution can be obtained by solving the variational problem in a finite dimensional subspace. Conditions for convergence and error estimates of the approximations are presented, both for interpolation problems and smoothing problems. The general theory is illustrated by specific examples including the “volume-matching” problem and the “one-sided thin plate spline”.

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Authors and Affiliations

  1. Department of Statistics, University of Chicago, 60637, Chicago, Illinois, USA

    Wing Hung Wong

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  1. Wing Hung Wong
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Additional information

This research is partially supported by the U.S. Army Contract No. DAAG 29-77-G-0207, and by NSF Grant No. MCS-8101836

Part of this paper is based on Chapters 2 and 3 of the author's Ph. D. thesis. The author would like to express his sincere thanks to Professor Grance Wahba and to two referees for many helpful comments

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Wong, W.H. On constrained multivariate splines and their approximations. Numer. Math. 43, 141–152 (1984). https://doi.org/10.1007/BF01389643

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  • DOI: https://doi.org/10.1007/BF01389643

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