On the construction of frames for Triebel-Lizorkin and Besov spaces

Kyriazis, George; Petrushev, Pencho

doi:10.1090/S0002-9939-05-08199-2

On the construction of frames for Triebel-Lizorkin and Besov spaces
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by George Kyriazis and Pencho Petrushev PDF

Proc. Amer. Math. Soc. 134 (2006), 1759-1770 Request permission

Abstract:

We present a general method for construction of frames $\{\psi _I\}_{I\in \mathcal {D}}$ for Triebel-Lizorkin and Besov spaces, whose nature can be prescribed. In particular, our method allows for constructing frames consisting of rational functions or more general functions which are linear combinations of a fixed (small) number of shifts and dilates of a single smooth and rapidly decaying function $\theta$ such as the Gaussian $\theta (x)=\exp (-|x|^2)$. We also study the boundedness and invertibility of the frame operator $Sf=\sum _{I\in \mathcal {D}} \langle {f,\psi _I}\rangle \psi _I$ on Triebel-Lizorkin and Besov spaces and give necessary and sufficient conditions for the dual system $\{S^{-1}\psi \}_{I\in \mathcal {D}}$ to be a frame as well.

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