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

By performing the change of variable χ ′ = χ 1 ,wegetχ = χ ′ a and d cosh χ =
a
dcoshχ ′ a = λ(a−1 ,χ ′ )d cosh χ ′ . The Fourier-Helgason coefficients become
〈E ξ,ν |D a ψ〉 =
∫ 2π ∫ ˜χ
From the cocycle property
we get
0
〈E ξ,ν |D a ψ〉 =
0
λ 1 2 (a −1 ,χ ′ )=
λ 1 2 (a, χ
′
a )ψ(χ ′ ,ϕ)E ξ,ν (χ ′ a ,ϕ)λ(a−1 ,χ ′ )sinhχ ′ dχ ′ dϕ. (72)
∫ 2π ∫ ˜χ
0
0
[
1
λ 1 2 (a, χ ′ a ) = a sinh χ a
sinh χ
] 1
2
cosh pχ , (73)
cosh pχ a
λ 1 2 (a −1 ,χ ′ ) ψ(χ ′ ,ϕ) E ξ,ν (χ ′ a,ϕ)sinhχ ′ dχ ′ dϕ. (74)
Then, we split (71) in three parts:
∫ ∞
0
∫ σ
(.)α(a)da = (.)α(a)da
0
} {{ }
∫ 1
∫
σ
∞
+ (.)α(a)da + (.)α(a)da
1
σ
} {{ } σ
} {{ }
I 1 I 2
I 3
. (75)
Let us focus on the first integral. Developing the Fourier-Helgason kernel E ξ,ν
in (74), we obtain :
I 1 =
× ∣
∫ σ
α(a)da×
0
∫ ˜χ ∫ 2π
0
0
dµ(χ ′ ,ϕ) λ 1 2 (a −1 ,χ ′ ) ψ(χ ′ )(coshχ ′ a − sinh χ ′ a cos ϕ) − 1 2 +iν∣ ∣ ∣
2
.
Using the explicit form of χ ′ a , we have for various involved quantities the
following asymptotic behaviors at small scale a ≈ 0:
cosh pχ a ∼ 1+o(a),
cosh χ a ∼ 1+o(a),
sinh χ a ∼ a sinh pχ + o(a),
p
(cosh χ ′ a − sinh χ ′ a cos ϕ) − 1 2 +iν ∼ 1 − (− 1 2 + iν)a cos ϕ.
p
So we have the approximation
∫ ˜χ ∫ 2π
0
0
∼ a √ 2p
∫ ˜χ
0
dµ(χ ′ ,ϕ) λ 1 2 (a −1 ,χ ′ ) ψ(χ ′ )(coshχ ′ a − sinh χ′ a cos ϕ)− 1 2 +iν
∫ 2π
0
dµ(χ ′ ,ϕ)
[ ] 1 ( sinh 2pχ
2
1 − (− 1 )
sinh χ
2 + iν)a p cos ϕ .
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