2007/06/25 by Daryl Geller, Geller, Daryl, Azita Mayeli +1 · 3 citations
Computer Science · Mathematics · #35P05 #42B20 #42C40 #58J35 #58J40 #Advanced Harmonic Analysis Research #Advanced Mathematical Modeling in Engineering #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #math.CA #math.FA #msc:35P05 #msc:42B20 #msc:42C40 #msc:58J35 #msc:58J40
paper · pdf · doi:10.48550/arxiv.0706.3642
60 pages, 7 figures
openalex publication_date 2007/06/25 · arxiv created 2008/11/27 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \bf M be a smooth compact oriented Riemannian manifold, and let Δ be the Laplace-Beltrami operator on \bf M. Say 0 ≠ f ∈ S(\RR+), and that f(0) = 0. For t > 0, let Kt(x,y) denote the kernel of f(t2 Δ). Suppose f satisfies Daubechies' criterion, and b > 0. For each j, write \bf M as a disjoint union of measurable sets Ej,k with diameter at most baj, and comparable to baj if baj is sufficiently small. Take xj,k ∈ Ej,k. We then show that the functions ϕj,k(x)=[μ(Ej,k)]1/2 Kaj(xj,k,x) form a frame for (I-P)L2(\bf M), for b sufficiently small (here P is the projection onto the constant functions). Moreover, we show that the ratio of the frame bounds approaches 1 nearly quadratically as the dilation parameter approaches 1, so that the frame quickly becomes nearly tight (for b sufficiently small). Moreover, based upon how well-localized a function F ∈ (I-P)L2 is in space and in frequency, we can describe which terms in the summation F ∼ SF = ∑j ∑k < F,ϕj,k > ϕj,k are so small that they can be neglected. If n=2 and \bf M is the torus or the sphere, and f(s)=se-s (the "Mexican hat" situation), we obtain two explicit approximate formulas for the ϕj,k, one to be used when t is large, and one to be used when t is small. Finally we explain in what sense the kernel Kt(x,y) should itself be regarded as a continuous wavelet on \bf M, and characterize the Hölder continuous functions on \bf M by the size of their continuous wavelet transforms, for Hölder exponents strictly between 0 and 1.