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Continuous wavelet transforms on \(n\)-dimensional spheres. (English) Zbl 1333.42053

Spherical wavelets on \(\mathbb{S}^n\) are derived from approximate identities and conditions are established under which the wavelets, at small scales, behave like Euclidean wavelets. Given a weight \(\alpha:\mathbb{R}_+\to\mathbb{R}_+\), a family \(\{\Psi_\rho\}_{\rho>0}\subset \mathcal{L}^2(\mathbb{S}^n)\) is called a bilinear spherical wavelet if it satisfies the admissibility conditions \(\sum_{\kappa=1}^{N(n,\ell)}|\alpha_\ell^\kappa|(\Psi_\rho)|^2\, \alpha(\rho)\, d\rho=N(n,\ell)\) for any \(\ell=0,1,\dots\) and \(\int_{\mathbb{S}^n}\bigl|\int_R^\infty \Psi_\rho \hat{\ast}\Psi_\rho(x\cdot y) \alpha(\rho)\, d\rho\bigr|d\sigma(y)\leq \text{const.}\) for some constant independent of \(R\) and \(x\in\mathbb{S}^n\). Here, \((f\hat{\ast}g)(x\cdot y)=\int_{\mathrm{SO}(n+1)} f(g^{-1} x)\, h(g^{-1} y)\, d\nu(g)\).
For an admissible spherical wavelet \(\{\Psi_\rho\}_{\rho>0}\) one defines \[ \mathcal{W}_\Psi f(\rho,g)=\frac{1}{\Sigma_n}\int_{\mathbb{S}^n}\overline{\psi_{\rho}(g^{-1} x)}\, f(x)\, d\sigma(x),\quad (\rho,g)\in \mathbb{R}_+\times \mathrm{SO}(n+1). \] One has the reconstruction formula \[ f=\int_0^\infty f \ast (\overline\Psi_\rho \hat{\ast} \Psi_\rho)\, \alpha(\rho)\, d\rho \] where \(\rho\) is the index of the wavelets. It is proved that \(\mathcal{W}_\Psi\) is unitary from \(\mathcal{L}^2(\mathbb{S}^n)\) to \(\mathcal{L}^2 (\mathbb{R}_+\times \mathrm{SO}(n+1))\) with inner product defined by integration against \(d\nu(g)\,\alpha(\rho) d\rho\). The small scale behavior is described as follows.
If \(\Psi_\rho=\sum_{\ell=0}^\infty \sum_{k\in \mathcal{M}_n(\ell)} a_\ell^k(\Psi_\rho) \, Y_\ell^k\) is the spherical harmonic decomposition of \(\Psi_\rho\) in which \(a_\ell^k(\Psi_\rho)=\frac{1}{\ell^{k_1-1}A_\ell^k} O(\psi_k(\ell\rho))\) as \(\rho\to 0\) with \(\psi_k\in \mathcal{L}^2(\mathbb{R}_+, t^{n-1}\, dt)\) a piecewise smooth function such that, for some \(c>0\) and \(\epsilon<1\) given, \(\rho^n\sum_{\ell=0}^{[c/\rho]} \ell^{n-1} \psi_k(\ell\rho)<\epsilon\) with \(k=(k_1,\dots,k_{n-1})\in\mathcal{M}_{n-1}(\ell)\) given when \(k_1\leq K\) and where \(\lim_{\rho\to 0} a_\ell^k(\Psi_\rho)=0\) when \(k_1\geq K\), then there is a square integrable function \(F:\mathbb{R}^n\to\mathbb{C}\) such that \(\lim_{\rho\to 0} \rho^n \Psi_\rho (S^{-1}(\rho\xi))=F(\xi)\) holds pointwise where \(S\) denotes the stereographic projection.
Wavelets are formulated in terms of approximate identities. Theorem 3.9 states that if a kernel \(\Phi_R\), \(R>0\), has Gegenbauer coefficients that are decreasing and differentiable in \(R\) and \(\lim_{R\to \infty}\widehat{\Phi}_R(\ell)=0\) \((\ell\in\mathbb{N})\) then the associated wavelet \(\Psi_\rho\) is defined by \(\widehat{\Psi}_\rho(\ell)=(-\frac{1}{\alpha(\rho)}\frac{d}{d\rho}|\widehat{\Phi}_\rho(\ell)|^2)^{1/2}\) for \(\ell\in 0,1,\dots\) and \(\rho>0\). In this case the family of functions \[ \Psi_\rho=\sum_{\ell=0}^\infty\sum_{\kappa=0}^{N(n,\ell)}(-\frac{1}{\alpha(\rho)}\frac{d}{d\rho}|\widehat{\Phi}_\rho(\ell)|^2)^{1/2}\frac{\lambda}{\lambda+\ell} w_\ell(\kappa) Y_\ell^\kappa(x) \] is a bilinear spherical wavelet. Linear wavelets are defined in requiring the admissibility sum condition only for \(\kappa=0\) and a reconstruction formula is provided for linear wavelets. More specific zonal linear wavelets are considered as are several specific examples of linear wavelets and comparisons are made with other known spherical wavelets.

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)

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