On the asymptotic distribution of oscillation points in rational approximation

. . ( ) , , (m) (m)m, n(m) ^* ) ( d(m) — r _{m, n(m)} ^*) )/ , .

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The paper was written during the second author's visit at the Mathematical Institute of the Hungarian Academy of Sciences.     

SimultaneousL 1 approximation of a compact set of real-valued functions

Summary In this paper the uniqueness results found in simultaneous Chebychev approximation are extended to simultaneousL ₁ approximation. In particular a sufficient condition to guarantee uniqueness of a best approximate to aL ₁ compact set is given.This paper is taken in part from a thesis to be submitted by M. P. Carroll in partial fulfillment of the requirements for the Ph. D. degree in the Department of Mathematics at Rensselaer Polytechnic Institute.     

Best approximation of functions specified with an error at a finite number of points

It is proved that for convex and centrally symmetric classes of functions a linear method is included among the best (in a definite sense) methods of approximation from values specified with an error at a finite number of points. For some of the simplest classes linear best methods are constructed and their error is estimated.     

Best approximation of analytic functions from information about their values at a finite number of points

For a class of bounded and analytic functions defined in a simply connected region we construct the best linear method of approximation with respect to information about the values of the function at some points of the region. We show it is unique. We obtain limiting relations for the lower bound of the norm of the error of the best method on an arbitrary compacta with connected complement where the lower bound is taken with respect to nodes from the region of analyticity.Translated from Matematicheskii Zametki, Vol. 19, No. 1, pp. 29–40, January, 1976.     

On the cone of a finite dimensional compact convex set at a point

Four equivalent conditions for a convex cone in a Euclidean space to be an F_σ-set are given. Our result answers in the negative a recent open problem posed by Tam [5], characterizes the barrier cone of a convex set, and also provides an alternative proof for the known characterizations of the inner aperture of a convex set as given by Brønsted [2] and Larman [3].     

The capacity of the rational preimage of a compact set

It is shown that a well-known expression for the capacity of the preimage of a compact set under a polynomial map remains valid in the case of a rational map, provided that the standard capacity of the preimage is replaced by its capacity in the external field determined by the poles in C of the rational function determining the map.     

Learning a function from noisy samples at a finite sparse set of points

In learning theory the goal is to reconstruct a function defined on some (typically high dimensional) domain Ω, when only noisy values of this function at a sparse, discrete subset ω⊂Ω are available.In this work we use Koksma–Hlawka type estimates to obtain deterministic bounds on the so-called generalization error. The resulting estimates show that the generalization error tends to zero when the noise in the measurements tends to zero and the number of sampling points tends to infinity sufficiently fast.     

Uniform approximation of continuous functions by rational functions

Summary Viene data la condizione necessaria e sufficiente perchè le funzioni razionali di una variabile, aventi poli di ordine prefissato in assegnati punti del piano complesso, costituiscano un sistema completo iu C^o (0, 1). A Bruno Finzi nel suo70 ^mo compleanno. This research has been sponsored in part by the Aerospace Research Laboratoires through the European Office of Aerospace Research, OAR, United States Air Force, under Grant EOOAR-69-0066. Entrata in Redazione il 22 aprile 1970.     

On rational approximation of algebraic functions

We construct a new scheme of approximation of any multivalued algebraic function f(z) by a sequence {r_n(z)}_n∈N of rational functions. The latter sequence is generated by a recurrence relation which is completely determined by the algebraic equation satisfied by f(z). Compared to the usual Padé approximation our scheme has a number of advantages, such as simple computational procedures that allow us to prove natural analogs of the Padé Conjecture and Nuttall's Conjecture for the sequence {r_n(z)}_n∈N in the complement CP¹?D_f, where D_f is the union of a finite number of segments of real algebraic curves and finitely many isolated points. In particular, our construction makes it possible to control the behavior of spurious poles and to describe the asymptotic ratio distribution of the family {r_n(z)}_n∈N. As an application we settle the so-called 3-conjecture of Egecioglu et al. dealing with a 4-term recursion related to a polynomial Riemann Hypothesis.     

Best uniform approximation to a class of rational functions

We explicitly determine the best uniform polynomial approximation to a class of rational functions of the form 1/²(x−c)+K(a,b,c,n)/(x−c) on [a,b] represented by their Chebyshev expansion, where a, b, and c are real numbers, n−1 denotes the degree of the best approximating polynomial, and K is a constant determined by a, b, c, and n. Our result is based on the explicit determination of a phase angle η in the representation of the approximation error by a trigonometric function. Moreover, we formulate an ansatz which offers a heuristic strategies to determine the best approximating polynomial to a function represented by its Chebyshev expansion. Combined with the phase angle method, this ansatz can be used to find the best uniform approximation to some more functions.     

Generic existence of a nonempty compact set of fixed points

Let X be a complete metric space, a set of continuous mappings from X into itself, endowed with a metric topology finer than the compact-open topology. Assuming that there exists a dense subset contained in such that for every mapping T in the set {x ε X: Tx = x} is nonempty, it is proved that most mappings (in the Baire category sense) in do have a nonempty compact set of fixed points. Some applications to α-nonexpansive operators, semiaccretive operators and differential equations in Banach spaces are derived.     

Behavior of alternation points in best rational approximation

The behavior of the equioscillation points (alternants) for the error in best uniform approximation on [–1, 1] by rational functions of degreen is investigated. In general, the points of the alternants need not be dense in [–1, 1], even when approximation by rational functions of degree (m, n) is considered and asymptoticallym/n 1. We show, however, that if more thanO(logn) poles of the approximants stay at a positive distance from [–1, 1], then asymptotic denseness holds, at least for a subsequence. Furthermore, we obtain stronger distribution results when n (0 < 1) poles stay away from [–1, 1]. In the special case when a Markoff function is approximated, the distribution of the equioscillation points is related to the asymptotics for the degree of approximation.The research of this author was supported, in part, by NSF grant DMS 920-3659.     

On the approximation by rational combinations of a class of functions

() [0,1] — {(n)} — , +. , f(x) [0,1] () , x ₁ ,x ₂ [0, 1], (₁)=(₂), f(x ₁ )=f(x ₂ ).     

ON THE INVERSE IMAGE SET OF RATIONAL FUNCTIONS

This article studies the inverse image of rational functions.Several theorems are obtained on the Julia set expressed by the inverse image,and a mistake is pointed out in H.Brolin' theorem incidentally.