Advances in Pure Mathematics
Vol.04 No.11(2014), Article ID:51915,8 pages
10.4236/apm.2014.411069
Some Results on Wavelet Frame Packets
Sana Khan, Mohammad Kalimuddin Ahmad
Department of Mathematics, Aligarh Muslim University, Aligarh, India
Email: sana17khan53@gmail.com, ahmad_kalimuddin@yahoo.co.in
Copyright © 2014 by authors and Scientific Research Publishing Inc.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/
Received 14 September 2014; revised 18 October 2014; accepted 29 October 2014
ABSTRACT
The aim of this paper is to study wavelet frame packets in which there are many frames. It is a generalization of wavelet packets. We derive few results on wavelet frame packets and have obtained the corresponding frame bounds.
Keywords:
Wavelet, Wavelet Packets, Frame Packets
1. Introduction
Let us consider an orthonormal wavelet of. The orthonormal wavelet bases
have a frequency localization which is proportional to
at the resolution level
. If we consider a bandlimited wavelet
(i.e.
is compactly supported), the measure of supp
is
times the measure of supp
, since
where. The wavelet bases have poor frequency localization when
is large. For some applications, it is more convenient to have orthonormal bases with better frequency localization. This will be provided by the wavelet packets.
The wavelet packets introduced by Coifman, Meyer and Wickerhauser [1] [2] played an important role in the applications of wavelet analysis. But the theory itself is worthy for further study. Some developments in the wavelet packet theory should be mentioned, for instance Shen [3] generalized the notion of univariate orthogonal wavelet packets to the case of multivariate wavelet packets. Chui and Li [4] generalized the concept of orthogonal wavelet packets to the case of nonorthogonal wavelet packets. Yang [5] constructed a-scale orthogonal multiwavelet packets which were more flexible in applications. In [6] , Chen and Cheng studied compactly supported orthogonal vector-valued wavelets and wavelet packets. Other notable generalizations are biorthogonal wavelet packets [7] and non-orthogonal wavelet packets with r-scaling functions [8] . For a nice exposition of wavelet packets of, see [9] .
The main tool used in the construction of wavelet packets is the splitting trick [10] . Let be an MRA of
with the corresponding scaling function
and the wavelet
. Let
be the corresponding wavelet subspaces
. In the construction of a wavelet from an MRA, the space
is split into two orthogonal components
and
, where
is the closure of the linear span of the func-
tions and
and
are the closure of the span of
and
respectively. Since
, we see that the above procedure splits the half integer translates of a function into the integer translates of two functions.
We can also choose to split which is the span of
. We
then have two functions whose translates will span the same space
. Repeating the splitting procedure
times, we get
functions whose integer translates alone span the space
. If we apply this to each
, then the resulting basis of
will give us a better frequency localization. This basis is called “wavelet packet basis”.
There are many orthonormal bases in the wavelet packets. Efficient algorithms for finding the best possible basis do exist; however for certain wavelet applications in signal analysis, frames are more suitable than orthonormal bases, due to the redundancy in frames. Therefore, it is worthwhile to generalize the construction of wavelet packets to wavelet frame packets in which there are many frames. The wavelet frame packets on was studied in [11] , and the frame packets on
were studied by Long and Chen in [12] [13] . Also, multiwavelet packets and frame packets of
were discussed in [14] .
Throughout the paper, the space of all square integrable functions on the real line will be denoted by and the inner product and Fourier transform of functions in
is given by
and
respectively. Also the norm of any in
will be denoted by
and the relationship be-
tween functions and their Fourier transform is defined by. For
, the
Fourier transform of
is in
and satisfies the Parseval identity
. Also, let
be the collection of almost everywhere (a.e.) bounded functions, i.e., functions bounded everywhere except on sets of (Lebesgue) measure zero and equipped with the norm
2. Wavelet Packets and Wavelet Frame Packets
Definition 1. A multiresolution analysis (MRA) consists of a sequence of closed subspaces,
of
and a function
, such that the following conditions hold:
1)
2) and
.
3).
4)
5) is an orthonormal basis for
.
The function is called the scaling function of the given MRA.
Suppose that generates a multiresolution analysis and that there exists some function
in
such that
is the orthogonal complement of
in
. Then
is called a basic wavelet relative to
.
If is a basic wavelet relative to
, then it is clear that the wavelet spaces
generated by
, satisfy the following properties:
6).
7).
8).
Since both the scaling function and the wavelet
are in
and
is generated by
, there exists two sequences
and
in
such that
(1)
(2)
for all. For the Haar basis, we have
(3)
(4)
Therefore, for the Haar basis, the scaling function and the wavelet function satisfy the following recurrence equation
(5)
(6)
Due to Coifman, Meyer and Wickerhauser [1] [2] , we have the following sequences of functions
(7)
(8)
where and
is the filter which satisfies the following properties
where is the Kronecker delta defined by
and
For in (7) and (8), we get
(9)
(10)
corresponds to our scaling function
and
corresponds to the wavelet
. If we increase
, we get the following structures
and so on. The functions, m = 2n or 2n + 1, n = 0, 1, 2,
are called “wavelet packets” scaling to the scaling function
. Thus, the family
is a generalization of the wavelet
Definition 2. The family,
is called a wavelet basis packet, where
is the oscillation parameter,
the scaling parameter and
the translation parameter.
We can also write. The family
constitutes wavelet frame packets if there are constants
and
,
such that
(11)
3. Main Results
Define,
,
and
Consider
and
Theorem 1. Let be the basic wavelet packets such that
and
Then constitutes wavelet frame packets with frame bounds
and
.
Proof. Let be the class of all those functions
such that
and
is compactly supported in
. By using the Parseval identity, we have
Since, , we have
Hence,
(12)
Let for
. Each
is compactly supported in
and belongs to
. If
is such a function,
which is -periodic and whose Fourier coefficients are
,
, then by Poisson sum formula we have,
Hence,
(13)
But the left side of (13) equals
(14)
It follows that
(15)
Applying (15) when in (12) we obtain
where,
In the expression for, the parameter
is a non-zero integer. For each such
there is a unique non- negative integer
and a unique odd integer
such that
. Therefore, we have
Thus,
(16)
for all. By using Schwarz’s inequality we have
By changing variables in the second integral and using the fact that, and applying Schwarz’s inequality for series we have
Hence,
These inequalities together with (16) give us
.
Since is dense in
, the above inequality holds for all
.
Theorem 2. The system,
is orthonormal if and only if
(17)
and
(18)
Proof. By using the Plancherel theorem we have
Thus, is orthonormal if and only if
a.e. The converse is immediate. Performing a change of variables, we see that
; this tells us that the system
is orthonormal for each fixed
when (17) is satisfied. The proof of condition (18) is similar.
Lemma 1. If is an orthonormal system, then
(19)
for all.
Proof. Let be the R.H.S of (19). We have to show that
for a.e.
. We first show that
and then that
; this will clearly give us (19). Using (18), with
replaced by
, we have
Replacing by
, we have
This shows that Now, we calculate
and show that
. Changing variables in the sum over
, we have
By using (17) and (18), we have
Theorem 3. Let be a sequence of wavelet frame packets with bounds
and
. Define
by
(20)
If the numbers satisfy the two conditions
(21)
(22)
then defined by (20) is a wavelet frame packet with bounds
and
.
Proof. Let. Then
(23)
By Cauchy-Schwarz inequality, we get
On solving the second term in the last product, we have
Thus,
By (23), we have
Thus,
Similarly, one can prove the upper frame condition.
References
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