Re: Orthogonal polynomials (was Chebyshv, etc.)

David C. Ullrich a écrit :
On Fri, 29 May 2009 23:36:32 +0100, Angus Rodgers
<twirlip@xxxxxxxxxxx> wrote:

On Fri, 29 May 2009 18:15:49 +0200, Denis Feldmann
<denis.feldmann.sansspam@xxxxxxx> wrote:

Checkiong Wikipedia, I realized there are many more mysteries on those. For instance , all those polynomials (of degree n) (like Legendre, Chebyshev, Hermite, etc.) have n real roots in the interval of integration. Any simple proof?
This general property of orthogonal polynomials is proved as
Theorem 12.2 in M. J. D. Powell, /Approximation Theory and
Methods/ - and presumably in many other places as well. The
proof is indeed short, but I'm way too tired to type it out
now. If someone doesn't provide a more accessible reference,
I might get round to typing it out tomorrow. It has something
to do with considering how often the polynomial changes sign
in the interval, multiplying it by another non-zero polynomial
which changes sign at the same points, so that the product is
non-negative, integrating this, and getting a non-zero scalar
product, so that the multiplying polynomial cannot be of lower
degree, because of the orthogonality properties, therefore the
number of sign changes is at least n, and because it cannot be
more than n, it is = n. Hope that will do as a rough sketch.

That's a proof. More formally, if (P_n) is a family of
polynomials, deg(P_n) = n, and the P_n are orthogonal
with respect to some weight on the interval I, then
P_n is orthogonal to every polynomial of degree less
than n. Hence P_n has at least (and hence exactly) n
zeroes on I, because if not then there exists a polyomial
Q with deg(Q) < n such that P_n and Q have the same
zeroes on I, and hence the inner product of P_n and Q
is non-zero.

Well, it is not exactly a proof, because you must first eliminate multiple roots of P_n. But yes, the idea is enough to let the details to the reader...




Zzzzz...

David C. Ullrich

"Understanding Godel isn't about following his formal proof. That would make a mockery of everything Godel was up to."
(John Jones, "My talk about Godel to the post-grads."
in sci.logic.)
.

Relevant Pages

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  • Re: Orthogonal polynomials (was Chebyshv, etc.)
    ... For instance, all those polynomials have n real roots in the interval of integration. ... multiplying it by another non-zero polynomial ... Then phi_k has exactly k real and distinct zeros in the open interval a < x < b. ... of the orthogonality properties of phi_k, r is not less than k. ...
    (sci.math)