In applied mathematics, vanishing moments of wavelets are crucial for the measurement of the local regularity of a signal. It is not as important to use a wavelet with a narrow frequency support. If the wavelet has n vanishing moments, the wavelet transform can be interpreted as a multiscale differential operator of order n. This yields a first relation between the differentiability of ƒ and its wavelet transform decay at fine scales.
The Lipschitz property approximates
f
with a polynomial
p
v
in the neighborhood of
v
:
f
(
t
)
=
p
v
(
t
)
+
ε
v
(
t
)
with
ε
v
(
t
)
≦
K
|
t
−
v
|
α
.
A wavelet transform estimates the exponent
α
by ignoring the polynomial
p
v
. For this purpose, we use a wavelet that has
n
>
α
vanishing moments:
∫
−
∞
∞
t
k
ψ
(
t
)
d
t
=
0
for
0
⩽
K
<
n
.
A wavelet with
n
vanishing moments is orthogonal to polynomials of degree
n
−
1
. Since
n
>
α
, the polynomial
p
v
has degree of at most
n
−
1
. With the change of variable
t
′
=
(
t
−
u
)
/
s
, we verify that:
W
p
v
(
u
,
s
)
=
∫
−
∞
∞
P
v
(
1
s
ψ
(
t
−
u
s
)
)
d
t
=
0.
Since
f
=
p
v
+
ε
v
,
W
f
(
u
,
s
)
=
W
ε
v
(
u
,
s
)
A wavelet with
n
vanishing moments can be written as the
n
th-order derivative of a function
θ
; the resulting wavelet transform is a multiscale differential operator. We suppose that
ψ
has a fast decay, which means that for any decay exponent
m
∈
N
there exists
C
m
such that:
∀
t
∈
R
,
|
ψ
(
t
)
|
≤
C
m
1
+
|
t
|
m
If
K
=
∫
−
∞
∞
θ
(
t
)
d
t
≠
0
, then the convolution
f
∗
θ
s
(
t
)
¯
can be interpreted as a weighted average of
f
with a kernel dilated by
s
. So
W
f
(
u
,
s
)
=
W
ε
v
(
u
,
s
)
proves that
W
f
(
u
,
s
)
is an
n
th-order derivative of an averaging of
f
over a domain proportional to
s
.
Since
θ
has a fast decay, one can verify that:
lim
s
→
0
1
s
θ
¯
s
=
K
δ
in the sense of the weak convergence. This means that for any
ϕ
that is continuous at
u
:
lim
s
→
0
ϕ
∗
1
s
θ
¯
s
(
u
)
=
K
ϕ
(
u
)
.
If
f
is
n
times continuously differentiable in the neighborhood of
u
, then
W
f
(
u
,
s
)
=
W
ε
v
(
u
,
s
)
implies that:
lim
s
→
0
W
f
(
u
,
s
)
s
n
+
1
/
2
=
lim
s
→
0
θ
¯
s
(
u
)
=
K
f
(
n
)
(
u
)
In particular, if
f
is
C
n
with a bounded
n
th-order derivative, then
|
W
f
(
u
,
s
)
|
=
O
(
s
n
+
1
/
2
)
. This is a first relation between the decay of
|
W
f
(
u
,
s
)
|
when
s
decreases and the uniform regularity of
f
.
A wavelet
ψ
with a fast decay has
n
vanishing moments if and only if there exists
θ
with a fast decay such that:
ψ
(
t
)
=
(
−
1
)
n
d
n
θ
(
t
)
d
t
n
As a consequence:
W
f
(
u
,
s
)
=
s
n
d
n
d
u
n
(
f
∗
θ
s
¯
)
(
u
)
with
θ
s
¯
(
t
)
=
s
−
1
/
2
θ
(
−
t
/
s
)
. Moreover,
ψ
has no more than
n
vanishing moments if and only if
∫
−
∞
∞
t
k
ψ
(
t
)
d
t
≠
0
.
