COMPOSITE WAVELET TRANSFORMS
function
f
on !Rn is defined by
Ff(e) =
f
f(x)
eix·~
dx,
}Rn
3
For 0
~
a b
~
oo, we write
J:
f(TJ)dll(TJ) to denote the integral of the form
f[a,b) f(TJ)dJ.t(TJ).
DEFINITION 2.1. Let
q
be a measurable function on !Rn satisfying the following
conditions:
(a) q
E
£1
n
Lr for some r
1;
(b) the least radial decreasing majorant of q is integrable, i.e.
ij(x)
=
sup lq(y)l
E
£1;
IYIIxl
(c)
JRn
q(x) dx =
1.
We denote
(2.1)
t
0,
and set
(2.2)
W f(x, t)
=
1
00
Qtrd(x)
d~-t(TJ),
where J.t is a finite Borel measure on
[0,
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.
The integral
(2.2)
is well-defined for any function
f
E
Lp, and
IIWf(·,t)IIP
~ 11~-tllllqlllllfllp.
where
11~-tll
=
Jro,oo) dllli(TJ). We will also consider a more general weighted transform
(2.3)
Waf(x, t)
=
1
00
Qtrd(x) e-at"'
d~-t(TJ),
where a
:2:
0 is a fixed parameter.
The choice of the kernel function q, the wavelet measure J.t, and the parameter
a
:2:
0 are in our disposal. This feature makes the new transform convenient in
applications.
2.2. Calderon's identity.
An
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
(2.4) J.t([O,oo))
=
0
and
1
00
llogTJidl~-ti(TJ)
oo.
Iff
E
Lp,
1 ~
p
~
oo1, and
1
00
1
c~-'
=
log-
d~-t(TJ),
0
1J
1
We remind the reader that
L
00
is interpreted as the space Co with the uniform convergence.
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