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