Uniform approximation by polynomials with integral coefficients I

Approximation by Polynomials with Coefficients

Uniform approximation by (quantum) polynomials

We show that quantum algorithms can be used to re-prove a classical theorem in approximation theory, Jackson’s Theorem, which gives a nearly-optimal quantitative version of Weierstrass’s Theorem on uniform approximation of continuous functions by polynomials. We provide two proofs, based respectively on quantum counting and on quantum phase estimation.

Uniform approximation by discrete least squares polynomials

We study uniform approximation of differentiable or analytic functions of one or several variables on a compact set K by a sequence of discrete least squares polynomials. In particular, if K satisfies a Markov inequality and we use point evaluations on standard discretization grids with the number of points growing polynomially in the degree, these polynomials provide nearly optimal approximant...

Uniform approximation and Bernstein polynomials with coefficients in the unit interval

This paper presents two main results. The first result pertains to uniform approximation with Bernstein polynomials. We show that, given a power-form polynomial g, we can obtain a Bernstein polynomial of degree m with coefficients that are as close as desired to the corresponding values of g evaluated at the points 0, 1 m , . . . , 1, provided that m is sufficiently large. The second result per...

Polynomials with Integral Coefficients, Equivalent to a given Polynomial

Let f(x0, . . . , xn) be a homogeneous polynomial with rational coefficients. The aim of this paper is to find a polynomial with integral coefficients F (x0, . . . , xn) which is “equivalent” to f and as “simple” as possible. The principal ingredient of the proof is to connect this question with the geometric invariant theory of polynomials. Applications to binary forms, class numbers, quadrati...