J.J Swetits

Best approximation by monotone functions

For 1 d p < co, let L, denote the Banach space of pth power Lebesgue integrable functions on the interval [0, l] with /I f IID = (lh 1 f / p)“p. Let M, EL, denote the set of non-decreasing functions. Then M, is a closed convex lattice. For 1 < p < co, each f E L, has a unique best approximation from M,, while, for p = 1, existence of a best approximation from M, follows from Proposition 4 of [6]. Recently, there has been interest in characterizing best L, approximations from M, [ 1, 2, 3, 41. For example, in [ 1 ] it is shown that iff E L, and if each point in [0, l] is a Lebesgue point off [7], then the best L, approximation to f from M, is unique and continuous. In each of the papers mentioned above, the approach taken was measure theoretic, and the arguments were necessarily complicated. The purpose of this paper is to approach the best approximation problem from a duality viewpoint. This leads to considerable simplification in the derivation of the results, and allows for the omission of the assumption that f E L ~.

On summability and positive linear operators

Abstract Quantitative estimates for approximation by positive linear operators are obtained with the use of a summability method which includes both convergence and almost convergence.

Approximation in L p [0,1] by n-convex functions

We characterize best L p approximation to feL p [0,1] from n-convex functions in L p [0,1], for 1≤p<∞ and for a positive integer n. This characterization is used to derive some additional properties of the best approximations.

On extremal sets and strong unicity constants for certain C∞ functions

Abstract For each f continuous on the interval I , let B n ( f ) denote the best uniform polynomial approximation of degree less than or equal to n . Let M n ( f ) denote the corresponding strong unicity constant. For a certain class of nonrational functions F , it is shown that there exist positive constants α and β and a natural number N such that αn ⩽ M n ( f ) ⩽ βn for n ⩾ N . The results of the present paper also provide concise estimates to the location of the extreme points of f − B n ( f ). The set F includes the functions f α ( x ) = e αx , α ≠ 0.

Uniform strong unicity for rational approximation

Abstract Let R n m denote the class of rational functions defined on a closed interval I with numerators in the class of polynomials of degree at most n and positive valued denominators in the class of polynomials of degree at most m . If f ϵ C ( I ) is normal, the well-known strong unicity theorem asserts that there is a smallest positive constant γ n , m ( f ) such that ∥ f − R ∥ ⩾ ∥ f − R f ∥ + γ n , m ( f )∥ R − R f ∥ for all R ϵ R n m , where R f is the best uniform approximation to f from R n m . In this paper, the dependence of γ n , m ( f ) on f is investigated. Sufficient conditions are given to insure that inf fϵΓ γ n , m ( f ) > 0, where Γ is a subset of C( I ). Necessity of these conditions is investigated and examples are given to show that known results for R n 0 do not directly extend to R n m for m > 0.

Quantitative estimates for Lp approximation with positive linear operators

Abstract Quantitative estimates for approximation with positive linear operators are derived. The results are in the same vein as recent results of Berens and DeVore. Two examples are provided.

Precise orders of strong unicity constants for a class of rational functions

Abstract Let R ⊆ C[∞-1, 1] denote a certain class of rational functions. For each f ϵ R, consider the polynomial of degree at most n that best approximates f in the uniform norm. The corresponding strong unicity constant is denoted by M n ( f ). Then there exist positive constants α and β, not depending on n , such that an ⩽ M n ( f )⩽ βn , n = 1,2,….

Orders of strong unicity constants

Abstract Given f ϵ C ( I ), the growth of the strong unicity constant M n ( f ) for changing dimension is considered. Under appropriate hypotheses it is shown that 2 n + 1 ⩽ M n ( f ) ⩽ βn 2 . Furthermore, relationships between certain Lebesgue constants and M n ( f ) are established.

Regularized Newton Methods for Minimization of Convex Quadratic Splines with Singular Hessians

A quadratic spline function on ℝ n is a differentiable piecewise quadratic function. Many convex quadratic programming problems can be reformulated as the problem of unconstrained minimization of convex quadratic splines. Therefore, it is important to investigate efficient algorithms for the unconstrained minimization of a convex quadratic spline. In this paper, for a convex quadratic spline f(x) that has a matrix representation and is bounded below, we show that one can find a global minimizer of f(x) in finitely many iterations even though the global minimizers of f(x) might form an unbounded set. The new idea is to use a regularized Newton direction when a Hessian matrix of f(x) is singular. Applications to quadratic programming are also included.

Unbounded functions and positive linear operators

Abstract The approximation of unbounded functions by positive linear operators under multiplier enlargement is investigated. It is shown that a very wide class of positive linear operators can be used to approximate functions with arbitrary growth on the real line. Estimates are given in terms of the usual quantities which appear in the Shisha-Mond theorem. Examples are provided.