Advances in Pure Mathematics
Vol.05 No.02(2015), Article ID:53586,8 pages
10.4236/apm.2015.52008
A Note on the Structure of Affine Subspaces of
Fengying Zhou, Xiaoyong Xu
School of Science, East China Institute of Technology, Nanchang, China
Email: zhoufengying@ecit.cn, xxy@ecit.cn
Copyright © 2015 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
Received 8 January 2015; accepted 26 January 2015; published 28 January 2015
ABSTRACT
This paper investigates the structure of general affine subspaces of. For a d × d expansive matrix A, it shows that every affine subspace can be decomposed as an orthogonal sum of spaces each of which is generated by dilating some shift invariant space in this affine subspace, and every non-zero and non-reducing affine subspace is the orthogonal direct sum of a reducing subspace and a purely non-reducing subspace, and every affine subspace is the orthogonal direct sum of at most three purely non-reducing subspaces when |detA| = 2.
Keywords:
Affine Subspace, Reducing Subspace, Shift Invariant Subspace, Orthogonal Sum
1. Introduction
Let A be a d × d expansive matrix. Define the dilation operator D and the shift operator Tk,
, by
respectively. It is easy to check that they are both unitary operators on. Given a closed subspace X of
, X is called a shift invariant subspace if
for every
; X is called a reducing subspace of
if
and
for every
; X is called an affine subspace of
if there exists an at most countable subset
of
such that
In this case, we say that
generates the affine subspace X. An affine subspace, which does not contain any non-zero reducing subspace, is called purely non-reducing. By Theorem 3.1 in [1] , a closed subspace X of
is an affine subspace if and only if
for some shift invariant subspace
M. Therefore an affine subspace X of
is a reducing subspace if and only if it is shift invariant. So far, the study of reducing subspaces has achieved fruitful results. The existence and construction of wavelet frames for an arbitrary reducing subspace can be seen in [2] -[7] . For one-dimensional case A = 2, Gu and Han investigated the existence of Parseval wavelet frames for singly generated affine subspaces in [8] and the structural properties of affine subspaces in [9] . For a given d × d expansive matrix A, Zhou and Li studied the construction of wavelet frames in the setting of finitely generated affine subspaces of
in [10] . For a general d × d expansive matrix A, this paper focuses on the structure of affine subspaces of
, which is a continuation of the literature [10] and has not been investigated yet.
2. Main Results
Lemma 1. Let X and Y be closed subspaces of a Hilbert space H and
be the orthogonal projection onto
. Then
1)
2)
Proof. 1) Obviously,. For the other direction, note that
then
So. Therefore,
. Thus 1) holds.
2) For, there exists some
such that
. So
, or
, which shows
due to the fact that
. For
, we have
and
. Thus for any
, there is g with
and
and
such that
. Consequently,
since. The proof is completed.
Lemma 2. Let
be a monotone sequence of subspaces in a Hilbert space
.
1) If
is increasing, then
2) If
is decreasing, then
Proof. We only prove 1) since 2) can be obtained similarly. Since
is increasing, the first equality is obvious and
If, then for any
, there exists
,
and
such that
and
. For such h, there is a unique sequence
and a unique
such that
for each
,
and
. This means that
The proof is completed.
Proposition 1. Suppose that X is an affine subspace of
with M being its generating shift invariant subspace. Then there exist a shift invariant subspace M1 in X and a reducing subspace Y of
contained
in X such that the length of M1 is no more than that of M and.
Proof. For each, define
Obviously,
for
and
. Let
. Similarly to the proof of Proposition 2.2 in
[10] , we know that Y is a reducing subspace. Now define. Then
by Lemma 2 and
Suppose that for some subset
such that
Since
is a shift invariant subspace, so is
. Thus for each
,
. Also note that
. Therefore, by Lemma 1,
which shows that M1 is a shift invariant subspace of length no more than the length of M. The proof is completed.
Proposition 2. Suppose that X is a non-zero affine subspace of
and Q is the maximal shift invariant subspace contained in X. Then the following hold:
1)
and
is a shift invariant subspace contained in X;
2)
if and only if X is purely non-reducing subspace of
.
Proof. 1): Obviously,
is shift invariant space since Q is shift invariant. So
due to the fact that Q is the maximal shift invariant subspace contained in X. Thus
is a shift invariant subspace contained in X.
2): By 1) and Lemma 2, it follows that
If X is purely non-reducing, then
since
is a reducing subspace. So
. Suppose
and X contains a reducing subspace Y. Next we only need to show
. Since Y is reducing, we have
and
, i.e.,
. Also note that
. Hence
. Thus for each
,
. Therefore
, which shows that
. The proof is completed.
Proposition 3. Let X be an affine subspace of, and define
. The
is the maximal shift invariant subspace contained in X.
Proof. We first show that
is shift invariant. For
and
, it follows that
Next we will show that
by contradiction. If there exists some
such that
, then
So. Therefore
, which implies that
since
. Consequently,
. This leads to a contradiction since
. Assume that M is a shift invariant subspace contained in X. Obviously
. Thus,
. So
. The result follows. The proof is completed.
Lemma 3. Let X and Y be affine subspaces of
with
. Let M and N be generating shift invariant subspaces for X and Y respectively. Then
is an affine subspace of
with
as a generating shift invariant subspace.
Proof. Since
and
, it follows that
The proof is completed.
Lemma 4. Assume
is a monotone sequence of subspaces in a Hilbert space
and give a subspace
satisfying
for each
. Then
Proof. Since
is a monotone sequence of subspaces and
,
, we have
is also a monotone sequence. Then the first equality follows by Lemma 2. For
, there exists some
such that
, namely
and
. Then
. Thus
. So
. For the other direction, without loss of generality, assume that
is increasing. By Lemma 2,
which shows that. The proof is completed.
Lemma 5. Let X be an affine subspace of
and Q be the maximal shift invariant subspace contained in X. Define
. Then the following hold:
1)
and
for
;
2),
if and only if X is a reducing subspace of
;
3),
is in any reducing subspace of
containing X.
Proof. 1): Note that we only need to show
and
. While
follows by Proposition 2. So
. Thus we have
2): Since Q is shift invariant and, it follows that
If X is a reducing subspace, then. By the definition of V, we have
. If
, then
, which shows that X is shift invariant. Thus X is a reducing subspace.
3): By 1) and 2), we have
for all j,
with
. Thus for each
,
. Therefore
. Let M be a reducing subspace containing X. Then
. So for each
,
. Hence
. The proof is completed.
Proposition 4. Let X and Y be affine subspaces of
satisfying
. Let Q and S be the maximal shift invariant subspaces contained in X and Y respectively. Define
. Then
is the maximal shift invariant subspace contained in
.
Proof. Let M be a shift invariant subspace contained in. By Lemma 3 and the maximality of S as a shift invariant subspace in Y, we have
. Note that
and
. Then
. So
. Hence
. Therefore
. The proof is completed.
Proposition 5. Let X and Y be affine subspaces of
satisfying
. Let Q and S be the maximal
shift invariant subspaces contained in X and Y respectively. Define. Then
is an affine subspace of
if and only if
.
Proof. According to Proposition 4,
is the maximal shift invariant subspace in
. If
is an affine subspace, then by Lemma 3,
is a generating shift invariant subspace for
, i.e.,
. Now suppose
. Since
and
by Lemma 5 for
, we have
for
. Thus by Lemma 4,
Write
and
. Then
. In fact,
Hence
due to the fact that
is equivalent to
for a given Hilbert space
with its two subspaces
and
. Also by Lemma 4, we have
So. The proof is completed.
Proposition 6. Let X and Y be two affine subspaces of
with
. Then the following holds.
1)
is affine if X is reducing;
2)
if Y is reducing and
is affine, where
and Q is the maximal shift invariant subspace in X.
Proof. 1): By Lemma 5,
with X being a reducing subspace. Then
is the maximal shift invariant subspace for
by Proposition 4. Now we only need to show that
. Note that
due to the facts that
and
. So by Lemma 1,
Observe that
since
is invariant under
for
. Therefore,
2): According to Proposition 5, it follows that. By Lemma 4,
which shows that, i.e.,
. Since
is contained in any reducing space containing X by Lemma 4,
. Consequently
. The proof is completed.
Theorem 7. Let X be an affine subspace of. Then the following holds.
1) There exist a shift invariant subspace M in X such that
for
with
, and
;
2) If X is a non-zero reducing subspace and, then there exist two purely non-reducing affine subspaces X1 and X2 such that
;
3) If X is non-zero and not reducing, then there exists a unique decomposition
with X1 be reducing and X2 being purely non-reducing;
4) If X is non-zero and, then X is the orthogonal direct sum of at most three purely non-reducing affine subspace.
Proof. 1): By Proposition 1, it follows that, where M1 is some shift invariant subspace in X and Y is a reducing subspace. If
, then the result follows. Otherwise, there is a
such that
is an orthonormal basis for Y. Let
and define
. Note that by the definition of M1 in the proof of Proposition 1, it follows that
for
with
. So
with
when
and
.
2): Let
be an orthonormal wavelet for X. Choose k,
such that
and
. Let
be a set of representatives of distinct cosets in
. Then
is a set of representatives of distinct cosets in
.
Indeed, for, clearly
if
. Now we consider the case
. Observe that
equals to
. Note that
. So
. Define two subsets
and
of X and two shift invariant subspaces P and M as follows:
Then
forms an orthonormal basis for P due to the fact that
is an orthonormal wavelet for X. The same to M. Define
Then. Next, we will show X1 is a purely non-reducing affine subspace. Write
Obviously Q is a shift invariant subspace contained in X1 and. According to Proposition 2, it suffices to show that Q is the maximal shift invariant subspace contained in X1. Also by Proposition 3, it is
enough to show, where
. Observe that for each
,
Then for each,
since
for all
. Therefore, for each
,
Hence
Thus. So X1 is a purely non-reducing affine subspace. Similarly to X2.
3): Let X be a non-reducing affine subspace of
and X1 be the maximal reducing subspace contained in X. Write
. Then X2 is affine by Proposition 6 and X2 is purely non-reducing since X1 is the maximal reducing subspace in X. Also note that the orthogonal complement of a reducing space within another reducing space is always reducing. Then the uniqueness follows.
4): 4) follows after 2) and 3). The proof is completed.
Acknowledgements
We thank the Editor and the referee for their comments. This work is funded by the National Natural Science Foundation of China (Grant No. 11326089), the Education Department Youth Science Foundation of Jiangxi Province (Grant No. GJJ14492) and PhD Research Startup Foundation of East China Institute of Technology (Grant No. DHBK2012205).
References
- Weiss, G. and Wilson, E.N. (2001) The Mathematical Theory of Wavelets. In: Byrnes, J.S., Ed., Twentieth Century Harmonic Analysis-A Celebration//Proceedings of the NATO Advanced Study Institute, Kluwer Academic Publishers, Dordrecht, 329-366.
- Dai, X., Diao, Y., Gu, Q. and Han, D. (2002) Frame Wavelets in Subspaces of
. Proceedings of the American Mathematical Society, 130, 3259-3267. http://dx.doi.org/10.1090/S0002-9939-02-06498-5
- Zhou, F.Y. and Li, Y.Z. (2010) Multivariate FMRAs and FMRA Frame Wavelets for Reducing Subspaces of
. Kyoto Journal of Mathematics, 50, 83-99. http://dx.doi.org/10.1215/0023608X-2009-006
- Dai, X., Diao, Y. and Gu, Q. (2002) Subspaces with Normalized Tight Frame Wavelets in
. Proceedings of the American Mathematical Society, 130, 1661-1667. http://dx.doi.org/10.1090/S0002-9939-01-06257-8
- Dai, X., Diao, Y., Gu, Q. and Han, D. (2003) The Existence of Subspace Wavelet Sets. Journal of Computational and Applied Mathematics, 155, 83-90. http://dx.doi.org/10.1016/S0377-0427(02)00893-2
- Lian, Q.F. and Li, Y.Z. (2007) Reducing Subspace Frame Multiresolution Analysis and Frame Wavelets. Communications on Pure and Applied Analysis, 6, 741-756. http://dx.doi.org/10.3934/cpaa.2007.6.741
- Li, Y.Z. and Zhou, F.Y. (2010) Affine and Quasi-Affine Dual Frames in Reducing Subspaces of
. Acta Mathematica Sinica (Chinese Edition), 53, 551-562.
- Gu, Q. and Han, D. (2009) Wavelet Frames for (Not Necessarily Reducing) Affine Subspaces. Applied and Computational Harmonic Analysis, 27, 47-54. http://dx.doi.org/10.1016/j.acha.2008.10.006
- Gu, Q. and Han, D. (2011) Wavelet Frames for (Not Necessarily Reducing) Affine Subspaces II: The Structure of Affine Subspaces. Journal of Functional Analysis, 260, 1615-1636. http://dx.doi.org/10.1016/j.jfa.2010.12.020
- Zhou, F.Y. and Li, Y.Z. (2013) A Note on Wavelet Frames for Affine Subspaces of
. Acta Mathematica Sinica (Chinese Edition), 33A, 89-97.