  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
Poincar\'e inequality which lead to heat kernels with small time Gaussian
bounds and H\"older 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.
