From: David Cantor <dgc@localhost.localdomain>
Subject: Re: Approximation by polynomials
Date: 20 Feb 2001 21:05:25 -0800
Newsgroups: sci.math.research
Summary: Approximating continuous functions using _integer_ polynomials

Non-trivial approximation by polynomials with integral coefficients on a compact
subset X of the real line is possible only if the transfinite diameter
of X is < 1.  If X is an interval [a,b] this means that b-a < 4.
When that is the case there are restrictions which depend upon the
complete sets of conjugate real algebraic integers in X.  Let T be the
union of all such sets.

If f is a continuous real function on [a,b] it can be approximated
arbitrarily well by polynomials with integral coefficients if and
only there exists a polynomial with integral coefficents g such that
f(t)=g(t) for all t in T. Note that is amounts to a finite number of
constraints.

If [a,b] = [-1.5,1.5] it is not hard to show that
T={-1,0,1,sqrt(2),-sqrt(2)}.  Thus, if for example f(-1)=1, f(0)=0,
f(1)=1, f(sqrt(20))=2 and f(-sqrt(2))=2, then f(t) agrees with t^2 on
T. Any such f can be approximated arbitrarily well by polynomials with
integral coefficients.

This and much more appears in an AMS publication by LeBaron Ferguson,
published many years ago.

David Cantor
Center for Communications Research--La Jolla
dgc@ccrwest.org