In his paper, Huybrechs considered a square-integrable function
f [member of] [L.sup.2](-1,1) that is not necessarily smooth or periodic.
We point out that, following , the Hilbert space can be taken to be given by a pair of square-integrable functions on the real line.
It consists of square-integrable functions defined on the real line (see [22, 23] for details).
Here we construct a chaotic linear operator by "lifting" 2B to the space [L.sub.2] of square-integrable functions (more precisely to the Hilbert space [L.sub.2](0, [pi]) of 2[pi]-periodic odd functions).
The Hilbert space [L.sub.2](0, [pi]) of square-integrable functions is isomorphic with [l.sub.2] (by the Riesz-Fischer theorem) and is a natural functional representation of the sequence space [l.sub.2].
The previous constructions allow us the definition of fractal bases of the space of square-integrable functions
on I .
[c.sub.p]--specific heat, J/kgK; [e.sub.RMS]--root mean square error, K; [h.sub.c]--thermal contact conductance, W/[m.sup.2]K; H--Hilbert space of square-integrable functions; I--maximum number of parameters; k--thermal conductivity, W/mK; L--length, m; N--number of sensors; q--heat flux, W/[m.sup.2]; S--objective function, [K.sup.2]; t--time, s; [t.sub.f]--final time, s; T--temperature, K; [T.sub.0]--constant temperatures at specimens' end, K; [T.sub.i]--initial temperature, K; Y--measured temperatures, K; x--Cartesian spatial coordinate.
We are after the function [h.sub.c](t) over the whole time domain (0, [t.sub.f]), with the assumption that [h.sub.c](t) belongs to the Hilbert space of square-integrable functions in the time domain --denoted as
A converse of the multi-band result for the classical case of continuous, square-integrable functions
is given in .
[L.sub.2]([Omega]) denotes the space of square-integrable functions
over [Omega] of [n.sub.sp] dimensions containing the function f having a finite [L.sub.2] norm of [[f]].