Compactly Supported Frames for Decomposition Spaces

“compactly” Supported Frames for Spaces of Distributions on the Ball

Frames are constructed on the unit ball Bd in Rd consisting of smooth functions with small shrinking supports. The new frames are designed so that they can be used for decomposition of weighted Triebel Lizorkin and Besov spaces on Bd with weight wμ(x) := (1 − |x|2)μ−1/2, μ half integer, μ ≥ 0.

Construction of compactly supported affine frames

Since the publication, less than ten years ago, of Mallat’s paper on Multiresoltion Analysis [Ma], and Daubechies’ paper on the construction of smooth compactly supported refinable functions [D], wavelets had gained enormous popularity in mathematics and in the application domains. It is sufficient to note that there are currently more than 10,000 subscribers to the monthly Wavelet Digest. At t...

Construction of Compactly Supported Tight Wavelets Frames

Compactly supported wavelet bases for Sobolev spaces

In this paper we investigate compactly supported wavelet bases for Sobolev spaces. Starting with a pair of compactly supported refinable functions φ and φ̃ in L2(R) satisfying a very mild condition, we provide a general principle for constructing a wavelet ψ such that the wavelets ψjk := 2j/2ψ(2j · − k) (j, k ∈ Z) form a Riesz basis for L2(R). If, in addition, φ lies in the Sobolev space H(R), t...

Compactly Supported Frames for Spaces of Distributions Associated with Non-negative Self-adjoint Operators

A small perturbation method is developed and deployed to the construction of frames with compactly supported elements of small shrinking supports for Besov and Triebel-Lizorkin spaces in the general setting of a doubling metric measure space in presence of a non-negative self-adjoint operator whose heat kernel has Gaussian localization and the Markov property. This allows, in particular, to dev...