COMPOSITE WAVELET TRANSFORMS
on !Rn is defined by
oo, we write
f(TJ)dll(TJ) to denote the integral of the form
DEFINITION 2.1. Let
be a measurable function on !Rn satisfying the following
Lr for some r
(b) the least radial decreasing majorant of q is integrable, i.e.
q(x) dx =
W f(x, t)
where J.t is a finite Borel measure on
oo). If J.t is a wavelet measure (i.e., 11 has
a certain number of vanishing moments and obeys suitable decay conditions) then
(2.2) will be called the composite wavelet transform of f. The function q will be
called a kernel function and Qt a kernel operator of the composite transform W.
is well-defined for any function
Jro,oo) dllli(TJ). We will also consider a more general weighted transform
0 is a fixed parameter.
The choice of the kernel function q, the wavelet measure J.t, and the parameter
0 are in our disposal. This feature makes the new transform convenient in
2.2. Calderon's identity.
analog of Calderon's reproducing formula for
Waf is given by the following theorem.
THEOREM 2.2. Let J.t be a finite Borel measure on [0, oo) satisfying
We remind the reader that
is interpreted as the space Co with the uniform convergence.