G. G. Lorentz

Approximation of functions

Possibility of Approximation: 1. Basic notions 2. Linear operators 3. Approximation theorems 4. The theorem of Stone 5. Notes Polynomials of Best Approximation: 1. Existence of polynomials of best approximation 2. Characterization of polynomials of best approximation 3. Applications of convexity 4. Chebyshev systems 5. Uniqueness of polynomials of best approximation 6. Chebyshevs theorem 7. Chebyshev polynomials 8. Approximation of some complex functions 9. Notes Properties of Polynomials and Moduli of Continuity: 1. Interpolation 2. Inequalities of Bernstein 3. The inequality of Markov 4. Growth of polynomials in the complex plane 5. Moduli of continuity 6. Moduli of smoothness 7. Classes of functions 8. Notes The Degree of Approximation by Trigonometric Polynomials: 1. Generalities 2. The theorem of Jackson 3. The degree of approximation of differentiable functions 4. Inverse theorems 5. Differentiable functions 6. Notes The Degree of Approximation by Algebraic Polynomials: 1. Preliminaries 2. The approximation theorems 3. Inequalities for the derivatives of polynomials 4. Inverse theorems 5. Approximation of analytic functions 6. Notes Approximation by Rational Functions. Functions of Several Variables: 1. Degree of rational approximation 2. Inverse theorems 3. Periodic functions of several variables 4. Approximation by algebraic polynomials 5. Notes Approximation by Linear Polynomial Operators: 1. Sums of de la Vallee-Poussin. Positive operators 2. The principle of uniform boundedness 3. Operators that preserve trigonometric polynomials 4. Trigonometric saturation classes 5. The saturation class of the Bernstein polynomials 6. Notes Approximation of Classes of Functions: 1. Introduction 2. Approximation in the space 3. The degree of approximation of the classes 4. Distance matrices 5. Approximation of the classes 6. Arbitrary moduli of continuity Approximation by operators 7. Analytic functions 8. Notes Widths: 1. Definitions and basic properties 2. Sets of continuous and differentiable functions 3. Widths of balls 4. Applications of theorem 2 5. Differential operators 6. Widths of the sets 7. Notes Entropy: 1. Entropy and capacity 2. Sets of continuous and differentiable functions 3. Entropy of classes of analytic functions 4. More general sets of analytic functions 5. Relations between entropy and widths 6. Notes Representation of Functions of Several Variables by Functions of One Variable: 1. The Theorem of Kolmogorov 2. The fundamental lemma 3. The completion of the proof 4. Functions not representable by superpositions 5. Notes Bibliography Index.

Approximation theory and functional analysis

Part 1 Research and survey articles: on Bernstein-Durrineyer polynomials with Jacob weights, H.Berens and Y.Xu an overview of wavelets, C.Chui a note on weak inequalities in Orlicz and Lorentz spaces, G.A.Edgar and L.Sucheston bivariate Birkhoff interpolation - a survey, R.A.Lorentz linear approximations of functions with several restricted derivatives, Y.Makovoz some characterizations theorems for measures associated with orthogonal polynomials on the unit circle, K.Pan and E.B.Saff box splines, cardinal series, and wavelets, S.D.Reimenschneider and Z.Shen some aspects of the subspace structure of infinite dimensional banach space, H.Rosenthal fairness and monotone curvature, J.Roulier et al projections on 2-dimensional spaces, N.Tomczak-Jacgermann real versus complex best rational approximation, R.S.Varga and A.Ruttan.

Relations between function spaces

With this norm, A(W) is a Banach space and even a Banach lattice, i.e., a vector lattice with the property that g I oo, and by a measure space (S, B, A). We have f EL( if

Metric Entropy, Widths, and Superpositions of Functions

Abstract : An introduction is given to some recent developments connected with properties of compact sets of continuous functions. Complete proofs are not given, but their main ideas are explained. (Author)

On a problem of additive number theory

Let A, B, denote sets of natural numbers. The counting function A(n) of A is the number of elements a CA which satisfy the inequality a (1-e)n/A(n), which holds for all large n.

Solvability Problems of Bivariate Interpolation I

This paper is devoted to bivariate interpolation. The problem is to find a polynomialP(x, y) whose values and the values of whose derivatives at given points match given data. Methods of Birkhoff interpolation are used throughout. We define interpolation matricesE, their regularity, their almost regularity, and finally the regularity of the pairE, Z for a given set of knotsZ. Many concrete examples and applications are possible.

Mathematics and Politics in the Soviet Union from 1928 to 1953

Abstract The paper describes the influence of politics on the life of Soviet mathematicians in Stalins era 1928–1953, years that witnessed the full unfolding of the dictators power. A few years following Stalins death are also covered. Various publications, private manuscripts, and recollections of my own experiences at the University of Leningrad served as sources. Leading themes include the administrative talent of Egorov, Lusins School, and the genius of Kolmogorov.

Lower bounds for the degree of approximation

is the optimal degree of approximation of 2W. In this paper we shall give simple methods which permit to find the order of magnitude of Dn(W) for several important classes ?1: for some classes of analytic functions (?6); for the unit ball AP+a of the space CP+a of functions with continuous derivatives of order p, which satisfy a Lipschitz condition with exponent a, 0 <a< 1 (?5). We also consider approximation of continuous functions (??2, 3) and give results about condensation of singularities (Theorems 2, 7). The main content of this paper consists of results which show that standard means (trigonometric approximation, series of orthogonal polynomials) give the best possible approximation, at least up to a bounded factor. Since estimates of Dn(21) from above follow from classical results [1; 5], we are interested in estimating Dn(2I) from below. Clearly, results of this type are the better, the smaller the norm used in the definition (1). This is why most of our theorems are for the LI-norm. Kolmogorov [2] (see also [6]) discussed D-n(2) in the L2-norm, and gave an asymptotic formula for it when 2 is the class of functionsf on an interval with f(P) bounded in the L2-norm. Originally the present paper was written to improve (for linear approximation) the

DISTRIBUTION OF ALTERNATION POINTS IN UNIFORM POLYNOMIAL APPROXIMATION

For a continuous function / on (0,1), we discuss the points where the polynomial Pn(x) of best uniform approximation deviates most from f(x), and the signs of the difference f(x) — Pn(x) alternate. We show that these points can be very irregularly distributed in (0,1), even if / is entire.

Bounds for polynomials with applications

Abstract For a polynomial having a non-constant upper bound on an interval, we derive upper bounds valid outside of that interval. Several applications are given. The paper arose from a desire to have a simpler proof of a result of one of us and to extend it to the complex plane.