Eli Passow

Chebyshev polynomials : from approximation theory to algebra and number theory

Definitions and Some Elementary Properties Extremal Properties Expansion of Functions in Series of Chebyshev Polynomials Iterative Properties and Some Remarks About the Graphs of the Tn Some Algebraic and Number Theoretic Properties of the Chebyshev Polynomials References Glossary of Symbols Index.

MONOTONE AND CONVEX SPLINE INTERPOLATION

For a set of monotone (and/or convex) data, we consider the possibility of finding a spline interpolant, of pre-determined smoothness, which is monotone (and/or convex). The investigation is carried out by constructing an auxiliary set of points and using the well-known monotonicity and convexity preserving properties of Bernstein polynomials. In § 3 we consider the problem of piecewise monotone interpolation.

Algorithms for Computing Shape Preserving Spline Interpolations to Data

Algorithms are presented for computing a smooth piecewise polynomial interpolation which preserves the monotonicity and/or convexity of the data.

Piecewise monotone spline interpolation

Abstract Let ( x i , y i ), i = 0, 1,…, k , be a set of points, with x 0 x 1 x k . We prove the existence of a spline function of specified deficiency, f ( x ), which satisfies f ( x i ) = y i , i = 0, 1,…, k , and which is monotone on each of the intervals [ x i − 1 , x i ], i = 1, 2,…, k .

Piecewise monotone polynomial approximation

Given a real function f satisfying a Lipschitz condition of order 1 on (a, b), there exists a sequence of approximating polynomials IP I such that the sequence En = |Pn - f| (sup norm) has order of magnitude I/n (D. Jackson). We investigate the possibility of selecting polynomials P having the same local n monotonicity as f without affecting the order of magnitude of the error. In particu- lar, we establish that if f has a finite number of maxima and minima on (a, b) and S is a closed subset of La, b) not containing any of the extreme points of f, then there is a sequence of polynomials P such that E has order of magnitude I/n and such that for n sufficiently large P and f have the same monotonicity at each point of S. The methods are classical.

Rational interpolation to | x | at the Chebyshev nodes

Recently the authors considered Newman-type rational interpolation to | x | induced by arbitrary sets of interpolation nodes and showed that under mild restrictions on the location of the interpolation nodes, the corresponding sequence of rational interpolants converges to | x |. In the present paper we consider the special case of the Chebyshev nodes which are known to be very efficient for polynomial interpolation. It is shown that, in contrast to the polynomial case, the approximation of | x | induced by rational interpolation at the Chebyshev nodes has the same order as rational interpolation at equidistant points.

Hermite-Birkhoff interpolation: A class of nonpoised matrices

has a unique solution for all cii . Then E is said to be poised. The problem is to find conditions which guarantee that E is or is not poised. Since a simple characterization of poised matrices is not currently known, attention has turned to the poisedness of classes of matrices. For example, De Vore et al. [2] considered three-row matrices with Hermite data (i.e., a single sequence of ones beginning in column 0, followed by a sequence of zeros) in rows 1 and 3, and just two ones in row 2. Lorentz [5] h as studied similar matrices. Partial solutions were obtained in these cases. In this paper we introduce a related class of matrices which we show are nonpoised.

Conditionally poised Birkhoff interpolation problems

Much attention has centered recently on Birkhoff interpolation, which can be described as follows. Let E = (e&Z: yCO be a matrix of zeros and ones, with exactly n -t1 ones. (For the sake of convenience, our notation will differ slightly from the usual.) Let x0 < x1 < ... x~+~ be interpolation nodes. Without loss of generality, it is possible to specify two of the nodes, and we set x, = I ) .Y(;+l = 1. Denote by P, the set of algebraic polynomials of degree < n. Let X =:= (.x1 , s2 ,..., x,.), and suppose that the system of equations