From: George Jones <gjones@uvi.edu>
Subject: Re: Orthogonal system for L^2(R)
Date: Tue, 19 Dec 2000 16:49:01 -0400
Newsgroups: sci.math
Summary: [missing]

In article <1pf8rkeidand@forum.mathforum.com>, Roger Gross wrote:

> The Hilbert space L^2(0,1) of square integrable functions on the unit
> interval has a complete orthogonal system consisting of the
> trigonometric functions cos(2pi*n*x) , sin(2pi*n*x) n=0,1,2,...
> is there a familiar complete orthogonal system for L^2(R), the Hilbert
> space of square integrable functions on the real line ?
> Thanks,
> Roger

L^2(R) is a separable Hilbert space, and thus has an infinite number
of countable complete orthogonal systems.

A concrete example of a countable complete orthogonal system is
{h_n (x) = exp(-x^2/2 * H_n (x) }, where the H_n (x) are the
Hermite polynomials.

Regards,
George
==============================================================================

From: Stephen Montgomery-Smith <stephen@math.missouri.edu>
Subject: Re: Orthogonal system for L^2(R)
Date: Tue, 19 Dec 2000 21:01:37 GMT
Newsgroups: sci.math

Roger Gross wrote:
> 
> The Hilbert space L^2(0,1) of square integrable functions on the unit
> interval has a complete orthogonal system consisting of the
> trigonometric functions cos(2pi*n*x) , sin(2pi*n*x) n=0,1,2,...
> is there a familiar complete orthogonal system for L^2(R), the Hilbert
> space of square integrable functions on the real line ?
> Thanks,
> Roger

Of course there are a bunch, but one is based on the Hermite polynomials.
The nth Hermite polynomial is something like

p_n(x) = c_n exp(x^2)  d^n/dx^n(exp(-x^2))

where c_n is an appropriate known constant.  Then the functions something
like

p_n(x) exp(-x^2/2)

form a complete o.n. system (which happen to be eigenvectors for the
Fourier Transform).

-- 
Stephen Montgomery-Smith
stephen@math.missouri.edu
http://www.math.missouri.edu/~stephen