IMI Interdisciplinary Mathematics InstituteCollege of Arts and Sciences

New bases for Triebel-Lizorkin and Besov spaces

A 1999 Preprint by G. Kyriazis and P. Petrushev

  • 1999:06
  • We give a new method for construction of unconditional bases for general classes of Triebel-Lizorkin and Besov spaces. These include the $L _ p$, $H _ p$, potential, and Sobolev spaces. The main feature of our method is that the character of the basis functions can be prescribed in a very general way. In particular, if $\Phi$ is any sufficiently smooth and rapidly decaying function, then our method construcs a basis whose elements are linear combinations of a fixed (small) number of shifts and dilates of the single function $\Phi$. Typical examples of such $\Phi$’s are the rational function $\Phi(\cdot)=(1+|\cdot|^2)^{-N}$ and the Gaussian function $\Phi(\cdot)=e^{-|\cdot|^2}$. This paper also shows how the new bases can be utilized in nonlinear approximation.

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