A remark on orthonormal bases of compactly supported wavelets in Triebel- Lizorkin spaces. The case \(0<p, q<\infty\)
Let \(\theta_ 0\in{\mathcal S}(R^ n)\) satisfy \(\theta_ 0(x)=1\) for \(| x|\leq 1\) and \(\theta(x)=0\) for \(| x|\geq 2\), and set \(\theta_ 1(x)=\theta_ 0(x/2)-\theta_ 0(x)\), \(\theta_ j(x)=\theta_ 0(2^{-j+1} x)\) for \(j\geq 1\). Then the non-homogeneous Triebel-Lizorkin space \(F^ s_{p,q}(R^ n)\) is defined as the subspace of tempered distributions for which the expression \(\left(\sum^ \infty_{j=0} 2^{jsq}|{\mathcal F}^{-1}(\theta_ j{\mathcal F}f)(x)|^ q\right)^{1/q}\) is in \(L^ p(R^ n)\). Furthermore, let a compactly supported \(m\)-regular wavelet basis be given, i.e., an orthonormal basis for \(L^ 2(R^ n)\) of the form \[ \{\phi_ 0(x-k), k\in Z\}\cup\{2^{jn/2}\phi_ i(2^ j x-k),\quad j, k\in Z,\;j\geq 1,\;i=1,\dots,2^ n-1\} \] for some functions \(\phi_ i\in C^ m\) with compact support. The main result of the paper is the following Theorem: Any \(m\)-regular compactly supported wavelet basis is an unconditional basis for \(F^ s_{p,q}(R^ n)\) for the range \(0< p,q<\infty\) and \(n \max(0,1/p-1,1/q-1)<s<m\). The theorem treats several cases in the range for \(p\), \(q\), \(s\) that are not covered in \textit{Y. Meyer's} book ``Ondelettes et opérateurs. I: Ondelettes'' (1990; Zbl 0694.41037). A similar theorem for the homogeneous Triebel-Lizorkin spaces has been obtained by the reviewer in Constructive theory of functions, Proc. Int. Conf., Varna/Bulg. 1987, 174-183 (1988; Zbl 0726.42015).
scientific article; zbMATH DE number 62239
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