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artur_steiner@yahoo.com
science forum beginner

Joined: 27 Sep 2005
Posts: 31

Posted: Tue Jul 18, 2006 1:20 am    Post subject: sequence of polynomials

Good evening,

I'd like some help, or a reference, to prove the following theorem:

Let (P_n) be a sequence of polynomials with real coefficients that
converges on R to a function P. If the sequence formed by the degrees
of the polynomials P_n is bounded, then P is a polynomial, too.

I tried using Bernstein's theorem, but it didn't lead to the desired
conclusion. Actually, I'm not sure what branch of Math is best for this
proof.

Thank you.

Artur
Robert B. Israel
science forum Guru

Joined: 24 Mar 2005
Posts: 2151

Posted: Tue Jul 18, 2006 2:20 am    Post subject: Re: sequence of polynomials

Artur <artur_steiner@yahoo.com> wrote:

 Quote: I'd like some help, or a reference, to prove the following theorem: Let (P_n) be a sequence of polynomials with real coefficients that converges on R to a function P. If the sequence formed by the degrees of the polynomials P_n is bounded, then P is a polynomial, too.

Suppose the polynomials all have degree <= M. Take any M+1 distinct
points x_0,...,x_M. For any y_0,...,y_M there is a unique polynomial
of degree <= M that takes values y_j at x_j, j=0...M, given by the
Lagrange Interpolation Formula. Thus there are linear functions
L_j of M+1 variables such that for any polynomial p(x) of degree
<= M, the coefficient of x^j in p(x) is L_j(p(x_0),...,p(x_M)).

Now the fact that P_n(x_i) -> P(x_i) for each i implies that
the coefficient of x^j in P_n, which is L_j(P_n(x_0),...,P_n(x_M)),
converges as n -> infty to L_j(P(x_0),...,P(x_M)). If
Q(x) = sum_{j=0}^M L_j(P(x_0),...,P(x_M)) x^j we then have
P_n(x) -> Q(x) as n -> infty for every x. In particular, by
uniqueness of limits Q(x) = P(x) for all real x.

Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia Vancouver, BC, Canada

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