## Abstract

Let A ∈ M_{n}(ℤ) be an expanding matrix with |det(A)| = q and let K = {k_{1} ⋯ k_{q}} ⊆ ℝ^{n} be a digit set. The set T =: T(A, K) = {∑_{i=1}^{∞} A^{-i} k_{ji} : k_{ji} ∈ K} ⊂ ℝ^{n} is called a self-affine tile if the Lebesgue measure of T is positive. In this note, we consider dilation equations of the form f(x) = ∑_{j=1}^{q} c_{j}f(Ax - k_{j}) with q = ∑_{j=1}^{q} c_{j}, c_{j} ∈ ℝ, and prove that this equation has a nontrivial L_{p} solution (1 ≤ p ≤ ∞) if and only if c_{j} = 1 ∀j ∈ {1,...,q} and T is a tile.

Original language | English |
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Pages (from-to) | 427-432 |

Number of pages | 6 |

Journal | Turkish Journal of Mathematics |

Volume | 25 |

Issue number | 3 |

Publication status | Published - 2001 |

Externally published | Yes |

## Keywords

- Dilation equtions
- Self-similar measures
- Tiles
- Wavelets

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