E. W. Cheney

Newton's method for convex programming and Tchebycheff approximation

w 1. Introduction. The rationale of Newtons method is exploited here in order to develop effective algorithms for solving the following general problem: given a convex continuous function F defined on a closed convex subset K of E,~, obtain (if such exists) a point x of K such that F(x)_<_F(y) for all y in K. The manifestation of Newtons method occurs when, in the course of computation, convex hypersurIaces are replaced by their support planes. The problems of infinite systems of linear inequalities and of infinite linear programming are subsumed by the above problem, as are certain Tchebycheff approximation problems for continuous functions on a metric compactum. In regard to the latter, special attention is devoted in w167 27--30 to the feasibility of replacing a continuum by a finite subset in such a way that a discrete approximation becomes an accurate substitute for the continuous approximation.

Strictly positive definite functions on spheres

A sufficient condition is given for the strict positive-definiteness of a real, continuous function on the m-dimensional sphere.

Minimal Projections on Hyperplanes in Sequence Spaces.

SummaryThe projection constants of hyperplanes in the classical sequence spaces (c0) and (l1) are determined, together with the projections of minimum norm.

Approximation from shift-invariant spaces by integral operators

We investigate approximation from shift-invariant spaces by using certain integral operators and discuss various applications of this approximation scheme. We assume that our integral operators commute with shift operators and that their kernel functions decay at a polynomial rate. We prove that the approximation order provided by such an integral operator is m if and only if the integral operator reproduces polynomials of degree up to m-1, where m is a positive integer. Using this result, we characterize the approximation order provided by a finitely generated shift-invariant space whose generators decay in a polynomial rate and have stable shifts. We also review some already well-studied approximation schemes such as projection, cardinal interpolation, and quasi-interpolation by considering them as special cases of integral operators.

Interpolation by piecewise-linear radial basis functions

Abstract In the two-dimensional plane, a set of nodes x 1 , x 2 , …, x n is given. It is desired to interpolate arbitrary data given at the nodes by a linear combination of the functions h i ( x ) = ∥ x − x i ∥. Here the norm is the l 1 -norm. For this purpose, one can employ the space PL of all continuous piecewise-linear functions on the rectangular grid generated by the nodes. Interpolation at the nodes by this larger space is quite easy. By adding an appropriate PL -function that vanishes on the nodes, we can obtain the linear combination of h 1 , h 2 , …, h n that interpolates the data. This algorithm is much more efficient than the straightforward method of simply solving the linear system of equations ∑ c j h j ( x i ) = d i .

Mean-square approximation by generalized rational functions

Abstract : The problem of numerical analysis to which this study is directed is that of determining an optimum approximation (in the least squares sense) to a given function f by a function of the form p/q, where p and q are confined to certain prescribed linear spaces. The analogous approximation problem employing the uniform norm has received much recent attention. To our knowledge no investigation of the present problem has appeared.

Quasi-interpolation with translates of a function having noncompact support

We establish a result related to a theorem of de Boor and Jia [1]. Their theorem, in turn, corrected and extended a result of Fix and Strang [5] concerning controlled approximation. In our result, the approximating functions are not required to have compact support, but satisfy instead conditions on their behavior at ∞. Our theorem includes some recent results of Jackson [6] and is closely related to the work of Buhmann [2].

Interpolation by periodic radial basis functions

Abstract On the unit circle S1, let d be the natural (geodesic) metric. We investigate the possibility of interpolating arbitrary data on a set of nodes yi ϵ S1 by means of a function of the form x ↦ ∑i = 1n cif(d(x, yi)). Here f is a function from [0, π] to R , and is subject to our choice. The interpolation matrix A having elements Aij = f(d(yj, yi)) is crucial to this problem. In the basic case, f(x) = x, we give necessary and sufficient conditions on the nodes for the invertibility of A. For equally-spaced nodes, we give nearly complete conditions on f for the invertibility of A.

A note on the operators arising in spline approximation

I. If, for each n, L,, is a linear projection of C[O, l] onto the subspace P,, of polynomials of degree G n then ,:L,li + ‘33. II. If, for each n, T, is the (nonlinear) metric projection of C[O, I] onto P, then JfTl n-‘) wheneverfg C[O, I].

Approximation theory VII

Approximation order without quasi-interpolants, C. de Bloor wavelets and signal analysis, C.K. Chui wavelet bases, approximation theory and subdivision schemes, A. Cohen approximation with convex rational functions, B. Gao, et al block structure and recursiveness in rational interpolation, M. Gutnecht multivariate approximation from the cardinal interpolation point of view, K. Jetter ridge functions, sugmoidal functions and neural networks, W. Light knot removal for spline curves and surfaces, T. Lyche approximation by algebraic polynomials, V. Totik.