Research Article | Open Access
Complete Invariance Property with respect to Homeomorphism over Frame Multiwavelet and Super-Wavelet Spaces
We discuss the complete invariance property with respect to homeomorphism (CIPH) over various sets of wavelets containing all orthonormal multiwavelets, all tight frame multiwavelets, all super-wavelets of length n, and all normalized tight super frame wavelets of length n.
A topological space is said to possess the complete invariance property (CIP) if each of its nonempty closed sets is the fixed point set, for some continuous self-map on . In case can be chosen to be a homeomorphism, the space is said to possess the complete invariance property with respect to homeomorphism (CIPH) . These notions have been extensively studied by Schirmer, Martin, Nadler, Oversteegen, Tymchatyn, Weiss, Chigogidge, and Hofmann. They studied the preservation of these properties under various topological operations such as products, cones and wedge products. They obtained various spaces with or without these properties.
Recently, Dubey and Vyas in  have studied the topological notion of the complete invariance property over the set , of all one-dimensional orthonormal wavelets on and certain subsets of . They noticed a free action of the unit circleon and obtained each orbit isometric to . They proved that the set of all one-dimensional orthonormal wavelets, the set of all MRA wavelets, and the set of all MSF wavelets on have the complete invariance property with respect to homeomorphism employing the following result of Martin : “A space has the CIPH if it satisfies the following conditions: (i) acts on freely. (ii) possesses a bounded metric such that each orbit is isometric to .”
In this paper, we study the complete invariance property with respect to homeomorphism over the spaces , containing all orthonormal multiwavelets on in-tuple form, , containing all tight frame multiwavelets on in -tuple form, is a super-wavelet of length for , and : is a normalized tight super frame wavelet of length . In case of the action of over , , and we obtain that the action is free but orbits are not isometric to . Observing this fact, we have proved that the result of Martin stated above is also true for orbits isometric to a circle of finite radius.
For a generic countable (or finite) index set such as , , , and , a collection of elements in a separable Hilbert spaceis called a frame if there exist constants and , , such that The optimal constants (maximal forand minimal for ) are called the frame bounds. is called a lower frame bound and is called an upper frame bound of the frame. The frame is called a tight frame if and is called normalized tight frame if . Any orthonormal basis in a Hilbert space is a normalized tight frame. Notice that, for a nonzero element of a frame in , the following inequality holds: This follows by noting that This shows that the elements of a frame need not be normal but they must have an upper bound.
Definition 1 (see ). Letbe an expansive matrix such that . Then a finite set is called an orthonormal multiwavelet if the collection is an orthonormal basis for , where for one uses the convention
If a multiwavelet consists of a single element , then we say that is a wavelet. By an expansive matrix , we mean a square matrix the moduli of whose eigenvalues are greater than.
If the collection is a normalized tight frame, then the set is called a normalized tight frame multiwavelet. Similarly, is called a tight frame multiwavelet when the above collection is a tight frame and a frame multiwavelet when the above collection is a frame.
The following result establishes a characterization of normalized tight frame multiwavelet.
Theorem 2 (see ). Suppose . Then the collection with a dilationis a normalized tight frame if and only if(i), for , where is the transpose of ,(ii), for and a.e..
In particularis a multiwavelet if and only if the above conditions hold and for all .
The Fourier transform ofis defined by where denotes the real inner product.
Sinceis a dense subset of, this definition extends uniquely to.
One of the methods of constructing orthonormal wavelets is based on multiresolution analysis which is a family of closed subspaces of satisfying certain properties.
Let be an orthonormal multiwavelet associated with a dilation.
Then describes the dimension functionfor, whereis the transpose of.
We have the following result analogous to that as in the case of one dimension.
Theorem 3 (see ). An orthonormal multiwaveletis an MRA multiwavelet if and only if, for a.e..
Definition 4 (see ). An MSF (minimally supported frequency) multiwavelet (of order L) is an orthonormal multiwavelet such that for some measurable set ,. An MSF multiwavelet of order 1 is simply referred to as an MSF wavelet.
The following theorem characterizes all MSF multiwavelets.
Theorem 5 (see ). A set such that for is an orthonormal multiwavelet with the dilation matrixif and only if
In , Han and Larson have introduced the notion of super-wavelet which has applications in signal processing, data compression, and image analysis.
Definition 6. Suppose that are normalized tight frame wavelets for . One will call the-tuple a super-wavelet of lengthif
is an orthonormal basis for (say,), where
Han and Larson in their memoirs  proved that, for each(can be), there is a super-wavelet of length.
Following is a characterization of a super-wavelet of length.
Theorem 7. Let . Then is a super-wavelet of lengthif and only if the following equations hold: (i)for a.e.(ii), for a.e. ,,(iii), for a.e. (iv), for a.e. ,.
Definition 8. A super-waveletis said to be an MRA super-wavelet if every is an MRA frame wavelet.
Definition 9 (see ). Suppose that . One will call the-tuple a normalized tight super frame wavelet of length if
is a normalized tight frame for .
For a self-continuous map on a topological space , Fix denotes the set of all fixed points of . A point is called a fixed point of if . In case is Hausdorff, Fix is a closed set.
From Brouwer’s fixed point theorem, it follows that Fixfor a self-continuous map on the disc is a nonempty closed set. The converse of this result was considered by Robbins who found it to be true . This is what led to the notion of the complete invariance property. Formally, we have the following.
Definition 10 (see ). A topological space is said to possess the complete invariance property (CIP) if each of its nonempty closed sets is Fix, for some self-continuous map on .
Definition 11 (see ). A topological space is said to possess the complete invariance property with respect to homeomorphism (CIPH) if each of its nonempty closed sets is the fixed point set, Fix, for some self-homeomorphism on .
Theorem 12 (see ). A space has the CIPH if it satisfies the following conditions: (i) acts on freely;(ii) possesses a bounded metric such that each orbit is isometric to the unit circle.
3. Frame Multiwavelet Spaces and the CIPH
From Theorem 12 it is clear that to examine the CIPH over a metric space we need a free action on having orbits isometric to the unit circle. This result does not provide any information about the CIPH over in case the radii of orbits are different from unity.
Below we modify the above result and show that if acts freely on a metric space and orbits are isometric to circles of finite radii, then has the CIPH.
Theorem 13. A space has the CIPH if it satisfies the following conditions:(i) acts on freely;(ii) possesses a bounded metric such that each orbit is isometric to , a circle of radius , where for some .
Proof. Let be a metric space with and let be the action satisfying conditions (i) and (ii). For a nonempty closed set in set and define by
Since if , it follows that . To see that is one-one suppose . Thenandmust lie on the same orbit isometric to the circle ,, and, for some real number , with .
Thus and hence, for some integer , By the triangle inequality applying over , , we have and so Thus the equation holds only for and hence .
Since is an orbit wise one-one map and a homeomorphism of into itself must be onto, it follows that is onto. In order to conclude that is a homeomorphism it suffices to show that is a closed mapping. For the remaining portion see the proof of Theorem 2.2 .
Let be an expansive matrix and is an integer. Then the space is a metric space with the natural metric defined by where and .
Theorem 14. The space has the CIPH.
Proof. The characterization of a normalized tight frame multiwavelet (Theorem 2) provides that if and , then, . For the collection forms a normalized tight frame for . This shows that cannot be a zero function in . Thus, the function defined by
is a free action of on .
For the continuity of at , we simply observe that where and The orbit of is given by which is isometric to , the circle of radius , via the map which sends to , where . Thus from Theorem 13 it follows that the space has the CIPH.
Corollary 15. If is an expansive matrix and is an integer, then the space has the CIPH.
Proof. Note that . The restriction of the action to is a free action. The orbit of is isometric to , the circle of radius , via the map sending to , where . Thus from Theorem 13 it follows that the space has the CIPH.
Remark 16. Let and . By noting that the dimension function of,, is equal to , it follows from Theorem 3 that is an MRA wavelet if and only if is an MRA wavelet. Also, we note that is an MSF wavelet if and only if is an MSF wavelet. Thus , , and are invariant sets in with respect to the action of topological group . The orbits of these invariant sets remain isometric to . Thus from Theorem 13 it follows that the spaces , , and have the CIPH.
In the case of tight frame, the frame bounds A and B are equal but need not be. After a renormalization, we can assume A = B = 1. If we denote then we have the following result.
Theorem 17. The space has the CIPH.
Theorem 18. Let Then the space has the CIPH.
Proof. Let ; that is, the collection
is a frame of .
Then for all , where and are frame bounds of the frame generated by .
Now, we show that is an element of . That is, for all .
Note that Hence we have Thus the map defined by is well defined and describes a free action of on .
The continuity of at follows by noting that where and
The orbit of is isometric to , the circle of radius , via the map sending to , where ; hence, the result is obtained.
4. Super-Wavelets and the CIPH
The concept of super-wavelets was first introduced and studied in . Due to its potential applications in multiplexing techniques such as time division multiple access and frequency division multiple access, super-wavelet has attracted the attentions of some mathematicians and engineering specialists. In this section we study the topological notion of the complete invariance property with respect to homeomorphism over the sets of super-wavelets and normalized tight super frame wavelets.
Theorem 19. Let be an integer. Consider the set defined by Then the space has the CIPH.
Proof. Let be an element of . From Theorem 7 it follows that remains in , where . Thus the map
is a free action.
The continuity of at follows by noting that where and .
The orbit of is given by which is isometric to , via the map which sends to . Thus from Theorem 12 it follows that the space has the CIPH.
Remark 20. If is an MRA super-wavelet of length for, then, for each , is also an MRA super-wavelet. Thus is an invariant set with respect to the action of . Orbits of these invariant sets are isometric to the unit circle. Thus from Theorem 12 it follows that the space has the CIPH.
Theorem 21. Let be an integer. Consider the set defined by : is a normalized tight super frame wavelet of length for . Then the space has the CIPH.
Proof. Let. Forwe have
This shows that remains in , where .
Thus the map defined by is a free action.
The continuity of at follows by noting that where and .
The orbit ofis isometric to ,, where . Thus from Theorem 13 it follows that the space has the CIPH.
For a , real expansive matrix , let and be the unitary operators ondefined by Then we have the following.
Definition 22 (see ). Suppose that are-dilation single normalized tight frame wavelets. One calls the -tuple an -dilation normalized tight super frame wavelet of lengthif
is an-dilation normalized tight frame for .
Thus from the above definition analogous result to Theorem 21 holds in case of higher dimension as well.
Remark 23. Consider the set Then we have similar results to the one-dimensional orthonormal wavelet case provided in . Let . Then the product space , the cylinder , where is the unit closed interval of the real line, the cone , and the suspension have the CIP.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
The author would like to thank Dr. Niraj K. Shukla, Discipline of Mathematics, Indian Institute of Technology Indore, Indore, India, for his help and suggestion in wavelet theory and the referee for his (her) careful reading of the paper and suggestions. This work was supported by CSIR grant, New Delhi, India.
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Copyright © 2014 Saurabh Chandra Maury. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.