Journal of Fourier Analysis and Applications

, Volume 18, Issue 5, pp 995–1066 | Cite as

Heat Kernel Generated Frames in the Setting of Dirichlet Spaces



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.


Heat kernel Gaussian bounds Functional calculus Sampling Frames Besov spaces 

Mathematics Subject Classification

58J35 42C15 43A85 46E35 


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Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Equipe AGM, CNRS-UMR 8088Université de Cergy-PontoiseCergy-PontoiseFrance
  2. 2.Laboratoire de Probabilités et Modèles Aléatoires, CNRS-UMR 7599Université Paris VI et Université Paris VIIParisFrance
  3. 3.Department of MathematicsUniversity of South CarolinaColumbiaUSA

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