Zvi Ziegler

Relative Chebyshev centers in normed linear spaces, part II

Abstract Let E be a normed linear space, A a bounded set in E , and G an arbitrary set in E . The relative Chebyshev center of A in G is the set of points in G best approximating A . We have obtained elsewhere general results characterizing the spaces in which the center reduces to a singleton in terms of structural properties related to uniform and strict convexity. In this paper, an analysis of the Chebyshev norm case, which falls outside the scope of the previous analysis, is presented.

Interlacing properties of the zeros of the error functions in best Lp-approximations

Let (u&L1 , +, and C,!J be given functions in C(i), where 1 is some fixed Gnitc interval, and let do be a finite nonatomic strictly positive measure on .L For p E [I, co], we denote by E,(#) and E,(G) the error functions in the best P-approximation to (band #, respectively, from [ul ,...+,I (--spanjul ,...,&]). For p < CO, the D-approximation is taken with respect to the measure da, For p = co, we shall consider the usual Tchebycheff (L”) approximation. The main result of this paper is the following theorem.

Korovkin shadows and Korovkin systems in C(S)-spaces

Abstract : The relationship between various types of Korovkin shadows is explored. An intrinsic characterization of Korovkin subspaces and Korovkin shadows is given. For some cases the detailed structure of the Korovkin shadows is presented.

Functions with Strictly Decreasing Distances from Increasing Tchebycheff Subspaces

Abstract Let { u i } i = 0 ∞ be a sequence of continuous functions on [0, 1] such that ( u 0 ,…, u k ) is a Tchebycheff system on [0, 1] for all k ⩾ 0 and let C ( u 0 ,…, u k ) denote the corresponding generalized convexity cone. It is proved that if f belongs to C ( u 0 ,…, u n − 1 ), then its distance from the linear space spanned by ( u 0 ,…, u n ) is strictly smaller than its distance from the linear space spanned by ( u 0 ,…, u n − 1 ). Other properties of the best approximants to such functions are also given. It is shown, by a general category argument, that no direct converse can exit. It is then established that if strict decrease of distances (or one of a number of other properties of the best approximants) holds for all subintervals of [0, 1], then f ϵ C ( u 0 ,…, u n − 1 ) for all of these.

Some Inequalities of Total Positivity in Pure and Applied Mathematics

We describe in this paper a spectrum of inequalities obtained through the use of total positivity methods mostly motivated by models of biological evolutionary processes, by problems originating in probability and statistical context, and by challenges of combinatoris.

Expansions of Generalized Completely Convex Functions

The concept of a generalized completely convex function is extended arid a unified presentation is developed for expanding such functions by Taylor–Lidstone series. It is shown that these expansions are in fact tantamount to representation theorems for the elements of the cone of generalized completely convex functions in terms of the extreme rays.

Approximating weak Chebyshev subspaces by Chebyshev subspaces

We examine to what extent finite-dimensional spaces defined on locally compact subsets of the line and possessing various weak Chebyshev properties (involving sign changes, zeros, alternation of best approximations, and peak points) can be uniformly approximated by a sequence of spaces having related properties.

Characterization of generalized convex functions by best L2-approximations

Let M be a normed linear space, and {Mn}1∞ a sequence of increasing finite dimensional subspaces, i.e., Mn ⊂ Mn + 1, for all n. For any element ƒ ϵ M, we obviously have d(ƒ, Mn) ⩾ d(ƒ, Mn + 1) , for all n, (1) where d(ƒ, Mk) is the distance, in the metric induced by the norm, from Mk to ƒ. In a recent paper [2], we discussed the space M = C[a,b] with the uniform norm and with Mn = [u0,…, un − 1], the linear subspace spanned by {ui}0n − 1, where {ui}0∞ is an infinite Tchebycheff system. We established there that the functions for which inequality (1), for a given n, is strict for all subintervals of [a, b] are precisely those that are convex with respect to (u0, u1,…, un − 1). The proof depended crucially on the alternance properties of the best approximants in the uniform norm. Somewhat surprisingly, analogous results are valid when the norm under consideration is the L2-norm. In fact, as we show in this paper, generalized convex functions play the same role in the L2-norm, for all continuous weights. Weaker results for the L2 case were obtained in [6].

FUNCTIONS OVER THE RESIDUE FIELD MODULO A PRIME

. In sectio 1n weprove that every functio inn one variable admits of a unique representationas a polynomia ol f degre ^p—1e in variable one . Explicit expressionsfor th coefficiente of a polynomias l representin a given functiog arenobtained. The main result osf th papee r are presented in sectio 2n , wherewe obtain necessar an sufficiend y t condition for th coefficiente s ofs apolynomial in order tha itt should represen at permutation. From theseconditions we derive some general conclusion the natur s abou oe f thtecoefficients o af polynomial representin a permutationg . In sectio 3n weapply the foregoing analysi tso the special circumstance Fs

Multiple node splines with boundary conditions: The fundamental theorem of algebra for monosplines and Gaussian quadrature formulae for splines

In this paper our main goal is to establish a “fundamental theorem of algebra” for monosplines having multiple knots and satisfying boundary conditions and to show the existence and uniqueness of multiple node Gaussian quadrature formulas for classes of splines where linear boundary functionals are included in the formulas. The known results on the “fundamental theorem” are due to the follow- ing individuals. For simple knots and multiple zeros, the results were proven by Karlin and Schumaker [4]. In the multiple knot and simple zero setting Micchelli [9] established the results. For multiple knots and multiple zeros the theorems were developed by Barrar and Loeb [ 11. For the simple knot, multiple zero, and the boundary condition case the results were derived by Karlin and Micchelli [IS]. With respect to the quadrature formulas the principal investigators were Karlin [3], Karlin and Pinkus [7], Melkman [8], Michelli and Pinkus [lo], and Karlin and Micchelli [S]. They developed the simple node case. Our point of departure is the paper of Micchelli and Pinkus. They exhibited the duality properties of the monospline and quadrature problems. Using this linkage and the topological methods developed in [l] we will establish similar results for multinle nodes and multiple zeros. 90