From: israel@math.ubc.ca (Robert Israel)
Subject: Re: measurable and integrable
Date: 25 Aug 1999 18:28:30 GMT
Newsgroups: sci.math
To: stevem@iea.com (Steve McGrew)
Keywords: Schauder bases in Banach spaces
In article <37c2a137.3470271@nntp.iea.com>,
stevem@iea.com (Steve McGrew) writes:
> On Tue, 24 Aug 1999 08:32:32 GMT, Robin Chapman
> wrote:
> >In article <37C23080.DA74978E@starlab.ifmo.ru>,
> > Alexander Belyakoff wrote:
> >> what's the difference beween measureable and integreable functions?
> >
> >Integrable implie measurable but not vice versa.
> Are there expressions -- like infinite series expressions with
> freely adjustable coefficients -- that specify large classes of
> measurable/integrable functions? For example, one that might work is
>
> Y = exp(-x^2)*P(x) where P(x) is any finite polynomial.
But it doesn't, because there are all sorts of integrable functions not
of this form.
> Is there an expression that would encompass *all* such
> functions?
I think what you're looking for are Schauder bases. A sequence {x_j} is
a Schauder basis of a Banach space X if every member of X can be written
in a unique way as sum_{j=1}^infinity c_j x_j for some sequence of scalars
c_j, the sum converging in the Banach-space norm. Not all separable Banach
spaces have these, but all the "classical" ones do. For example, in
L_1[0,1] (the integrable functions on the interval [0,1]) one Schauder
basis consists of the Haar functions:
h_1(t) = 1
h_j(t) = {1 for (2m-2) 2^(-k-1) <= t < (2m-1) 2^(-k-1)
{-1 for (2m-1) 2^(-k-1) <= t < (2m) 2^(-k-1)
{0 otherwise
where j = 2^k+m, 1 <= m <= 2^k
For L_1(R), the integrable functions on the real line, you could take
integer translates of the Haar functions: the doubly-indexed sequence
h_j(t+i), j = 1,2,..., i any integer, where h_j(t) = 0 for t < 0 or t > 1.
Robert Israel israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2