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sequence of polynomials
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artur_steiner@yahoo.com
science forum beginner


Joined: 27 Sep 2005
Posts: 31

PostPosted: Tue Jul 18, 2006 1:20 am    Post subject: sequence of polynomials Reply with quote

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
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Robert B. Israel
science forum Guru


Joined: 24 Mar 2005
Posts: 2151

PostPosted: Tue Jul 18, 2006 2:20 am    Post subject: Re: sequence of polynomials Reply with quote

In article <1153185651.994776.316270@i42g2000cwa.googlegroups.com>,
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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