Equivalence of A-approximate continuity for self-adjoint expansive linear maps
Let A : R d → R d , d ⩾ 1 , be an expansive linear map. The notion of A -approximate continuity was recently used to give a characterization of scaling functions in a multiresolution analysis (MRA). The definition of A -approximate continuity at a point x – or, equivalently, the definition of the fa...
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Published in | Linear algebra and its applications Vol. 429; no. 7; pp. 1504 - 1521 |
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Main Authors | , |
Format | Journal Article |
Language | English |
Published |
New York, NY
Elsevier Inc
01.10.2008
Elsevier Science |
Subjects | |
Online Access | Get full text |
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Summary: | Let
A
:
R
d
→
R
d
,
d
⩾
1
, be an expansive linear map. The notion of
A
-approximate continuity was recently used to give a characterization of scaling functions in a multiresolution analysis (MRA). The definition of
A
-approximate continuity at a point
x
– or, equivalently, the definition of the family of sets having
x
as point of
A
-density – depend on the expansive linear map
A
. The aim of the present paper is to characterize those self-adjoint expansive linear maps
A
1
,
A
2
:
R
d
→
R
d
for which the respective concepts of
A
μ
-approximate continuity
(
μ
=
1
,
2
)
coincide. These we apply to analyze the equivalence among dilation matrices for a construction of systems of MRA. In particular, we give a full description for the equivalence class of the dyadic dilation matrix among all self-adjoint expansive maps. If the so-called “four exponentials conjecture” of algebraic number theory holds true, then a similar full description follows even for general self-adjoint expansive linear maps, too. |
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ISSN: | 0024-3795 1873-1856 |
DOI: | 10.1016/j.laa.2008.04.028 |