On the Lp Solutions of Dilation Equations

İbrahim KIRAT

Article Details

On the Lp Solutions of Dilation Equations

Authors:

İbrahim KIRAT

Publisher:

TUBITAK

Journal:

Turkish Journal of Mathematics 2001 Vol.25 Issue 3, pp.427-432

Document Type:

Article

Publication Date:

2/2001

Document Language:

Turkish

Document ID (Org No.):

2000749064

Frequency:

Bimonthly

eISSN:

1303-6149

pISSN:

1300-0098

Keywords:

Abstract:

 

Let  A \in Mn ({\Bbb Z}) be an expanding matrix with  |  {\det (A)} | = q  and let K =  {k1 \cdots kq} \subseteq {\Bbb  R}n be a digit set.  The set \cal T =:\cal T(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 f(x) = \sumj=1q cj f(Ax- kj) with q=\sumj=1q  {cj}, cj\in {\Bbb R}, and prove that this equation has a  nontrivial Lp solution (1\leq p \leq \infty) if and only if  cj=1  \forall j\in {1,...,q} and \cal T is a tile.

 

Let  A \in Mn ({\Bbb Z}) be an expanding matrix with  |  {\det (A)} | = q  and let K =  {k1 \cdots kq} \subseteq {\Bbb  R}n be a digit set.  The set \cal T =:\cal T(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 f(x) = \sumj=1q cj f(Ax- kj) with q=\sumj=1q  {cj}, cj\in {\Bbb R}, and prove that this equation has a  nontrivial Lp solution (1\leq p \leq \infty) if and only if  cj=1  \forall j\in {1,...,q} and \cal T is a tile.

Article Details

On the Lp Solutions of Dilation Equations

Authors:

İbrahim KIRAT

Publisher:

TUBITAK

Journal:

Turkish Journal of Mathematics 2001 Vol.25 Issue 3, pp.427-432

Document Type:

Article

Publication Date:

2/2001

Document Language:

Turkish

Document ID (Org No.):

2000749064

Frequency:

Bimonthly

eISSN:

1303-6149

pISSN:

1300-0098

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

Issues

Article Details

Authors: İbrahim KIRAT

Publisher: TUBITAK

Journal: Turkish Journal of Mathematics 2001 Vol.25 Issue 3, pp.427-432

Document Type: Article

Publication Date: 2/2001

Document Language: Turkish

Document ID (Org No.): 2000749064

Frequency: Bimonthly

eISSN: 1303-6149

pISSN: 1300-0098

Issues

Authors:

İbrahim KIRAT

Publisher:

TUBITAK

Journal:

Turkish Journal of Mathematics 2001 Vol.25 Issue 3, pp.427-432

Document Type:

Article

Publication Date:

2/2001

Document Language:

Turkish

Document ID (Org No.):

2000749064

Frequency:

Bimonthly

eISSN:

1303-6149

pISSN:

1300-0098