Heat Kernel Generated Frames in the Setting of Dirichlet Spaces
By T. Coulhon, Gerard Kerkyacharian, Pencho Petrushev. Journal of Fourier Analysis and Applications, 2012
112 citations recorded by the source corpus.
Wavelet bases and frames consisting of band limited functions of nearly exponential localization on ℝd are a powerful tool in harmonic analysis by making various spaces of functions and distributions more accessible for study and utilization, and providing sparse representation of natural function spaces (e.g. Besov spaces) on ℝd. Such frames are also available on the sphere and in more general homogeneous spaces, on the interval and ball. The purpose of this article is to develop band limited well-localized frames in the general setting of Dirichlet spaces with doubling measure and a local scale-invariant Poincaré inequality which lead to heat kernels with small time Gaussian bounds and Hölder continuity. As an application of this construction, band limited frames are developed in the context of Lie groups or homogeneous spaces with polynomial volume growth, complete Riemannian manifolds with Ricci curvature bounded from below and satisfying the volume doubling property, and other settings. The new frames are used for decomposition of Besov spaces in this general setting.
Subfields: Applied Mathematics, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/ca23abbc58b210addfad0f178a8fbea450d7aaf7
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In Applied Mathematics
- Gerard Kerkyacharian
- The Anh Bui
- H. Bui
- Dominique Picard
- Xuan Thinh Duong
- I. Castillo
- Dachun Yang
- Lixin Yan
In Mathematical Physics
- H. Bui
- The Anh Bui
- Gerard Kerkyacharian
- Xuan Thinh Duong
- Dominique Picard
- I. Castillo
- Lixin Yan
- T. Coulhon
Citations in the corpus
- WEIGHTED BESOV AND TRIEBEL–LIZORKIN SPACES ASSOCIATED WITH OPERATORS AND APPLICATIONS (2020, background): “(iv) Recently, under the assumption that L is a nonnegative self-adjoint operator satisfying Gaussian upper bounds, Hölder continuity and Markov semigroup properties, the frame decompositions of Besov and Triebel–Lizorkin spaces associated with L with full range of indices were studied in [26, 36, 5”
- Thomas Bayes’ walk on manifolds (2012, methodology): “One can easily prove (see [9]) that under the assumptions above, necessarily there exists positive constants C1, C ′ 1 such that C′ 1 |B(x, √ t)| ≤ Pt(x, x) ≤ C1 |B(x, √ t)| (11)”