From: Vladimir Drobot <drobot@mathcs.sjsu.edu>
Subject: Re: Approximation by polynomials
Date: Tue, 13 Feb 2001 13:40:18 -0800
Newsgroups: sci.math.research
Summary: Approximate continuous functions on an interval by _integral_ polynomials?

There is a book devoted to the subject: Ferguson, Approximation by
polynomials with integral coefficients, AMS Surveys, vol 17, 1980. The
names connected with the subject are Gelfand (Gelfand, with 'a', not
Gelfond, and Hewitt) The situation is a bit complicated, the ability to
approximate on an interval [a, b] depends on a and b. Some of it is easy
to see, if [a, b] contains an integer x, and f(x) = 0.5, you cannot
approxiamate f. But, surprisingly, it also depends on the length of the
interval [a, b]

Vladimir Drobot


Bob Katz wrote:

> If the real function f(x) is continuous on the interval [0,1] then by
> the well known Stone-Weierstrass theorem we can approximate f(x)
> uniformly by polynomials with real coefficients.
>
> In general it is not possible to approximate f(x) uniformly by
> polynomials with integer coefficients because for such polynomial
> p(x), the numbers p(0) and p(1) are integers .
>
> But let me ask:
>
> If f(0) and f(1) are integers is such approximation possible ?
> i.e for every epsilon we can find polynomial with integer coefficients
> p(x) such that |p(x)-f(x)| <= epsilon
> uniformly on [0,1] .
>
> Thanks,
> Bob