İbrahim KIRAT

On the $L^p$ solutions of dilation equations

ISSN: 1300-0098
Yayın Aralığı: 6
Yayıncı: TÜBİTAK

Let A $in$ Mn($Bbb{Z}$) be an expanding matrix with |det(A)| = q and let K = {$k_1...k_q$} $subseteq Bbb{R}^n$ be a digit set. The set $tau =: tau(A, K) = {sum_{i=1}^{infty}A^{-i}{k_j}_i : {k_j}_i in K} subset Bbb{R}^n$ is called a self-affine tile if the Lebesgue measure of $tau$ is positive. In this note, we consider dilation equations of the form $f(x) = sum_{j=1}^{q}c_jf(Ax - k_j)$ with $q = sum_{j=1}^{q}c_j, c_j in Bbb{R}$, and prove that this equation has a nontrivial $L^p$ solution ($1leq p leq infty$ ) if and only if $c_j = 1 forall j in$ {1,...,q} and $tau$ is a tile.

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