Wavelets on Smooth Manifolds and Applications

1. Wavelets on Smooth Manifolds and Applications • M denotes a smoothorientedcompactRiemannianmanifold and ListheLaplace-Beltramioperator on M . • Spectral Theorem forManifolds. Letf be a nonzeroboundedfunctiondefined on thespecrum of L . Thenf(L) is a boundedoperatordefined on L²(M) with a kernelK defined onM x M. • Theorem [4]. Iff isSchwartz and f(0)=0, then K iswell-localized and and defines a continuouswaveletforL²(M). • Theorem [3]. K canbediscretizedintoawaveletframeprovided MAMEBIA, IBB

2. Wavelets on Smooth Manifolds and Applications Application to theStudy of Distribution Spaces • Theorem [2].Besov spaces are characterized through the knowledge of size of the wavelet frame coefficients. More precisely, a distribution belongs to a Besov space if and only if its frame coefficients belong to the associated coefficients Besov space: where MAMEBIA, IBB

3. Wavelets on Smooth Manifolds and Applications Application to Analyzing of SphericalData • Generalized mexicanneedlets are generated wavelets on the sphere by the kernel of operators associated to f(𝜉) = 𝜉pf0(𝜉), where f0is a schwartz function on (0, ∞), t>0, and p is an integral number. • Theorem [1]. Mexicanneedlets and their higher orders have Gaussian decay at each scale. • Theorem [1]. Theneedletscoefficientsareasymptoticallyuncorrelated at smallscales. • Mexican needlets do not oscillate for higher orders, so they can be implemented directly on the sphere, which is desirable if there is missing data (such as the “sky cut” of the CMB, see figure 1). MAMEBIA, IBB

4. Wavelets on Smooth Manifolds and Applications • Moreover, due to the well-localization property of mexicanneedlets, they are robust in the presence of partial observed spherical dates. Figure 1 The map is due to Frode Hansen and the data are from WMPA. MAMEBIA, IBB

5. Wavelets on Smooth Manifolds and Applications References [1] Mayeli, A., Asymptotic uncorrelation for generalized mexicanneedlets, submitted 2008. [2] Geller, D., Mayeli, A., Besov spaces and frames on compact manifolds, submitted 2008. [3] Geller, D., Mayeli, A., Nearly tight frames and space-frequency analysis on compact manifolds, to appear in Math. Z., 2008. [4] Geller, D., Mayeli, A., Continuous wavelets on compact manifolds, to appear in Math. Z., 2008. MAMEBIA, IBB