Smooth Orthogonal Projections on Riemannian Manifold

Abstract

We construct a decomposition of the identity operator on a Riemannian manifold M as a sum of smooth orthogonal projections subordinate to an open cover of M . This extends a decomposition of the real line by smooth orthogonal projection due to Coifman, Meyer [9] and Auscher, Weiss, Wickerhauser [3], and a similar decomposition when M is the sphere by the first two authors [4].

Parseval wavelet frames on Riemannian manifold

“…The theory of Triebel-Lizorkin and Besov spaces on such manifolds was further developed by Triebel [39], Skrzypczak [35], and Große and Schneider [21]. Our boundedness result is an extension of analogous result for Sobolev spaces shown in [3]. A prototype of this result is due Triebel [39] who showed the boundedness of composition with a global diffeomorphism on Triebel-Lizorkin spaces on R d .…”

“…

We construct Parseval wavelet frames in L 2 (M ) for a general Riemannian manifold M and we show the existence of wavelet unconditional frames in L p (M ) for 1 < p < ∞. This is made possible thanks to smooth orthogonal projection decomposition of the identity operator on L 2 (M ), which was recently proven by the authors in [3]. We also show a characterization of Triebel-Lizorkin F s p,q (M ) and Besov B s p,q (M ) spaces on compact manifolds in terms of magnitudes of coefficients of Parseval wavelet frames.

…”

“…In this paper we improve upon these results by showing the existence of smooth Parseval wavelet frames on arbitrary Riemannian manifold M. Hence, we eliminate compactness assumption on M needed in the work of Geller et al [18,19,20] or Ricci curvature assumptions and volume doubling property needed in [12], and at the same time improve the construction of nearly tight frames to that of Parseval (tight) wavelet frames on M. This construction is made possible thanks to smooth orthogonal projection decomposition of the identity operator on M, which is an operator analogue of omnipresent smooth partition of unity subordinate to an open cover U of M, recently shown by the authors in [3]. Our smooth orthogonal decomposition leads naturally to a decomposition of L 2 (M) as orthogonal subspaces consisting of functions localized on elements of an open and precompact cover U.…”

“…This construction extends to L p spaces and yields unconditional dual wavelet frames in L p (M) for the entire range 1 < p < ∞. This is made possible by the extension of the above mentioned result in [3] which yields a decomposition of the identity operator I on L p (M) as a sum of smooth projections P U , which are mutually disjoint…”

Multiresolution analysis and adaptive estimation on a sphere using stereographic wavelets

“…So if one can construct a family of kernels such that they satisfy conditions of Theorem 2.5, one obtains a method of minimax rate of convergence for the L 2 -loss, i.e., n −2s/(2s+d) without the logarithmic factor on the manifold M . The first step is done in [5] i.e., a smooth orthogonal decomposition of identity in L 2 (M ) is constructed.…”