Multivariate interpolation at arbitrary points made simple

The concrete method of surface spline interpolation is closely connected with the classical problem of minimizing a Sobolev seminorm under interpolatory constraints; the intrinsic structure of surface splines is accordingly that of a multivariate extension of natural splines. The proper abstract setting is a Hilbert function space whose reproducing kernel involves no functions more complicated than logarithms and is easily coded. Convenient representation formulas are given, as also a practical multivariate extension of the Peano kernel theorem. Owing to the numerical stability of Cholesky factorization of positive definite symmetric matrices, the whole construction process of a surface spline can be described as a recursive algorithm, the data relative to the various interpolation points being exploited in sequence.

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Résumé La méthode concrète d'interpolation par surfaces-spline est étroitement liée au problème classique de la minimisation d'une semi-norme de Soboleff sous des contraintes d'interpolation; la structure intrinsèque des surfaces-spline est dès lors celle d'une extension multivariée des fonctions-spline naturelles. Le cadre abstrait adéquat est un espace fonctionnel hilbertien dont le noyau reproduisant ne fait pas intervenir de fonctions plus compliquées que des logarithmes et est aisé à programmer. Des formules commodes de représentation sont données, ainsi qu'une extension multivariée d'intérêt pratique du théorème du noyau de Peano. Grâce à la stabilité numérique de la factorisation de Cholesky des matrices symétriques définies positives, la construction d'une surface-spline peut se faire en exploitant point après point les données d'interpolation.

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Dedicated to Professor E. Stiefel     

Convergence rates for lacunary trigonometric interpolation

Conditions for the solvability of lacunary trigonometric interpolation have been given by Cavaretta, Sharma, and Varga. When these conditions are satisfied, linear operators are defined on the space of continuous periodic functions. In this paper, the saturation rate and class of many of these operators is determined.     

Some remarks on lacunary interpolation in the roots of unity

The object of this note is to consider the problem of obtaining the explicit representations for polynomials of interpolation in the (0,2,3) case as explained in the introduction. We also show that Dini-Lipschitz condition suffices for the convergence problem, both in this and in the general result of Kis. This paper was completed while the author was at the University of Chicago under Air Force Grant AF-AFOSR-62-118 in the summer of 1962–63. The author is grateful to Professor A. Zygmund and to Professor Turán for several useful suggestions.     

Order regularity of two-node Birkhoff interpolation with lacunary polynomials

In this short work we study the existence and uniqueness of solution for some Birkhoff interpolation problems with lacunary polynomials. First we solve the one-node problem; next we solve the two-node problem in the restricted case where one of the nodes is null.     

A problem of P. Turán (on the mean convergence of lacunary interpolation

Acta Mathematica Hungarica -     

Distribution of interpolation points

We show that interpolation to a function, analytic on a compact setE in the complex plane, can yield maximal convergence only if a subsequence of the interpolation points converges to the equilibrium distribution onE in the weak sense. Furthermore, we will derive a converse theorem for the case when the measure associated with the interpolation points converges to a measure onE, which may be different from the equilibrium measure.     

Mixed interpolation methods with arbitrary nodes

Previous work on interpolation by linear combinations of the form aC(x) + bS(x) + ∑_i=0ⁿ⁻²α_ixⁱ, where C and S are given functions and the coefficients a, b, and {α_j} are determined by the interpolation conditions, was restricted to uniformly spaced interpolation nodes. Here we derive both Newtonian and Lagrangian formulae for the interpolant for arbitrarily chosen distinct nodes. In the Newtonian form the interpolating function is expressed as the sum of the interpolating polynomial based on the given nodes and two correction terms involving an auxiliary function for which a recurrence relation is obtained. Each canonical function for the Lagrangian form may be expressed as a product of the corresponding Lagrange polynomial and a function which depends on divided differences of C(x) and S(x).     

A note on correctors with an arbitrary number of nonstep points

In this note maximal order,k step correctors with one nonstep point for the solution ofy=f(x,y),y(x ₀)=y ₀, introduced by Gragg and Stetter [1] are extended to an arbitrary numbers of nonstep points. These correctors have order 2k + 2s, are proved stable fork8,s2, and unstable for largek.     

Newton interpolation at Leja points

The Newton form is a convenient representation for interpolation polynomials. Its sensitivity to perturbations depends on the distribution and ordering of the interpolation points. The present paper bounds the growth of the condition number of the Newton form when the interpolation points are Leja points for fairly general compact sets K in the complex plane. Because the Leja points are defined recursively, they are attractive to use with the Newton form. If K is an interval, then the Leja points are distributed roughly like Chebyshev points. Our investigation of the Newton form defined by interpolation at Leja points suggests an ordering scheme for arbitrary interpolation points.Research supported in part by NSF under Grant DMS-8704196 and by U.S. Air Force Grant AFSOR-87-0102.On leave from University of Kentucky, Department of Mathematics, Lexington, KY 40506, U.S.A.