Continuous Wavelet Transform on the Hyperboloid - Université de ...

get the more elaborate expression
λ(a, χ) = dcoshχ 1/a
dcoshχ = cosh χ
. (46)
a
√1+a 2 −2 sinh 2 χ
Thus, the action of the dilation operator on the function is
D a ψ(x) ≡ ψ a (x) =λ 1 2 (a, χ)ψ(d
−1
a x) =λ 1 2 (a, χ)ψ(x 1
a
) (47)
with x a ≡ (χ a ,ϕ) ∈ H 2 + and it reads
ψ a (x) = √ 1 a
sinh χ 1/a
sinh χ
cosh pχ
ψ(x 1 ).
cosh pχ 1/a
a
One easily checks using (45) that D a is unitary in L 2 (H 2 +).
Finally, the hyperbolic wavelet function can be written as
ψ a,g (x) =U g D a ψ(x) =U g ψ a (x).
Accordingly, the hyperbolic continuous wavelet transform of a signal (function)
f ∈ L 2 (H+ 2 ) is defined as:
W f (a, g)=〈ψ a,g |f〉 (48)
∫
= [U g D a ψ](x)f(x)dµ(x) (49)
H+
2
∫
= ψ a (g −1 x)f(x)dµ(x) (50)
H 2 +
where x ≡ (χ, ϕ) ∈ H 2 + and g ∈ SO 0 (1, 2).
In the next section, we show how this expression can be conveniently interpreted
and studied as a hyperbolic convolution.
6.1 Convolutions on H 2
Since H+ 2 is a homogeneous space of SO 0 (1, 2), one can easily define a convolution.
Indeed, let f ∈ L 2 (H+ 2 )ands ∈ L1 (H+ 2 ), their hyperbolic convolution
is the function of g ∈ SO 0 (1, 2) defined as
∫
(f ∗ s)(g) =
H 2 +
f(g −1 x)s(x)dµ(x). (51)
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