Nonuniqueness of simultaneous approximation by algebraic polynomials

Simultaneous Approximation of Polynomials

Let Pd denote the family of all polynomials of degree at most d in one variable x, with real coefficients. A sequence of positive numbers x1 ≤ x2 ≤ . . . is called Pd-controlling if there exist y1, y2, . . . ∈ R such that for every polynomial p ∈ Pd there exists an index i with |p(xi)−yi| ≤ 1. We settle an problem of Makai and Pach (1983) by showing that x1 ≤ x2 ≤ . . . is Pd-controlling if and...

Simultaneous Approximation and Algebraic Independence

We establish several new measures of simultaneous algebraic approximations for families of complex numbers (θ1, . . . , θn) related to the classical exponential and elliptic functions. These measures are completely explicit in terms of the degree and height of the algebraic approximations. In some instances, they imply that the fieldQ(θ1, . . . , θn) has transcendance degree≥2 overQ. This appro...

Simultaneous Approximation of Zero by Values of Integral Polynomials

Uniform and Pointwise Shape Preserving Approximation by Algebraic Polynomials

We survey developments, over the last thirty years, in the theory of Shape Preserving Approximation (SPA) by algebraic polynomials on a finite interval. In this article, “shape” refers to (finitely many changes of) monotonicity, convexity, or q-monotonicity of a function. It is rather well known that it is possible to approximate a function by algebraic polynomials that preserve its shape (i.e....

Effective simultaneous approximation of complex numbers by conjugate algebraic integers

We study effectively the simultaneous approximation of n − 1 different complex numbers by conjugate algebraic integers of degree n over Z( √ −1). This is a refinement of a result of Motzkin [2] (see also [3], p. 50) who has no estimate for the remaining conjugate. If the n−1 different complex numbers lie symmetrically about the real axis, then Z( √ −1) can be replaced by Z. In Section 1 we prov...