Analysis of \(\mu_{R,D}\)-orthogonality in affine iterated function systems
Let \(R\in M_n(\mathbb{Z})\) be an expanding integer matrix, \(D\subseteq\mathbb{Z}^n\) a finite subset and \(\mu_{R,D}\) a corresponding self-affine measure. The first main result in this paper, Theorem 2.3, concerns \(\mu_{R,D}\)-orthogonality and compatible pair conditions. This is some refined version of an earlier result [\textit{J.-L. Li}, J. Funct. Anal. 244, No. 2, 628--638 (2007; Zbl 1119.42014)] in a special case, but with the essential advantage to omit some seemingly artificial condition (already conjectured before). In Theorem~2.5 another conjecture, cf. [\textit{I. Łaba} and \textit{Y. Wang}, J. Funct. Anal. 193, No. 2, 409--420 (2002; Zbl 1016.28009)] about the equality of probability weights in case of a self-affine measure associated with an IFS can be verified in case of \(|D|=2\). Finally, in Section~3 the generalized plane Sierpiński gasket corresponding to \[ R=\begin{bmatrix} 2 & 1 \\ 0 & 2 \end{bmatrix}\qquad\text{and}\qquad D=\left\{\begin{pmatrix} 0 \\ 0 \end{pmatrix}, \begin{pmatrix} 1 \\ 0 \end{pmatrix}, \begin{pmatrix} 0 \\ 1 \end{pmatrix}\right\} \] is studied. In Theorem~3.3 the author shows that then for \(\mu_{R,D}\) there are at most 3 mutually orthogonal exponential functions in \(L^2(\mu_{R,D})\) and the number 3 is best.
scientific article; zbMATH DE number 5549551
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