On the Lp Solutions of Dilation Equations

Authors : İbrahim KIRAT

Pages : 427-432

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Publication Date : 0000-00-00

Article Type : Research Paper

Abstract :Let A \in Mn insert ignore into journalissuearticles values({\Bbb Z}); be an expanding matrix with | {\det insert ignore into journalissuearticles values(A);} | = q and let K = {k1 \cdots kq} \subseteq {\Bbb R}n be a digit set. The set \cal T =:\cal Tinsert ignore into journalissuearticles values(A,K); = {\sumi=1\infty A-i kji : kji \in K} \subset {\Bbb R}n is called a {\it self-affine tile} if the Lebesgue measure of \cal T is positive. In this note, we consider dilation equations of the form finsert ignore into journalissuearticles values(x); = \sumj=1q cj finsert ignore into journalissuearticles values(Ax- kj); with q=\sumj=1q {cj}, cj\in {\Bbb R}, and prove that this equation has a nontrivial Lp solution insert ignore into journalissuearticles values(1\leq p \leq \infty); if and only if cj=1 \forall j\in {1,...,q} and \cal T is a tile.
Keywords : Dilation equtions, tiles, wavelets, self similar measures 433 Zülfigar AKDOĞAN GOP Üniversitesi, Fen Edebiyat Fakültesi, Tokat TURKEY Abdullah MAĞDEN Atatürk Üniversitesi, Fen Edebiyat Fakültesi, Erzurum TURKEY Some Characteriza