Complete Invariance Property with respect to Homeomorphism over Frame Multiwavelet and Super-Wavelet Spaces

A topological space X 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 f on X [1]. In case f can be chosen to be a homeomorphism, the space is said to possess the complete invariance property with respect to homeomorphism (CIPH) [2]. 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 [3] have studied the topological notion of the complete invariance property over the set W, of all one-dimensional orthonormal wavelets on R and certain subsets of W. They noticed a free action of the unit circle S onW and obtained each orbit isometric to S. 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 R have the complete invariance property with respect to homeomorphism employing the following result of Martin [2]: “A spaceX has the CIPH if it satisfies the following conditions: (i) S acts on X freely. (ii) X possesses a bounded metric such that each orbit is isometric to S.” In this paper, we study the complete invariance property with respect to homeomorphism over the spaces W ⊂ ∏ 1≤j≤L L 2 (R), containing all orthonormal multiwavelets on R in L-tuple form, W T ⊂ ∏ 1≤j≤L L 2 (R), containing all tight frame multiwavelets on R in L-tuple form, SW n = {(η 1 , . . . , η n ) : (η 1 , . . . , η n ) is a superwavelet of length n for L(R)⊕n}, and SWNT n = {(η 1 , . . . , η n ) :


Introduction
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  [1].In case  can be chosen to be a homeomorphism, the space is said to possess the complete invariance property with respect to homeomorphism (CIPH) [2].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 [3] have studied the topological notion of the complete invariance property over the set W, of all one-dimensional orthonormal wavelets on R and certain subsets of W. They noticed a free action of the unit circle  1 on W and obtained each orbit isometric to  1 .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 R have the complete invariance property with respect to homeomorphism employing the following result of Martin [2]: "A space  has the CIPH if it satisfies the following conditions: (i)  1 acts on  freely.(ii)  possesses a bounded metric such that each orbit is isometric to  1 ." In this paper, we study the complete invariance property with respect to homeomorphism over the spaces  ⊂ ∏ 1≤≤  2 (R  ), containing all orthonormal multiwavelets on R  in -tuple form,   ⊂ ∏ 1≤≤  2 (R  ), containing all tight frame multiwavelets on R  in -tuple form, SW  = {( 1 , . . .,   ) : ( 1 , . . .,   ) is a superwavelet of length  for  2 (R) ⊕ }, and SW NT  = {( 1 , . . .,   ) : ( 1 , . . .,   ) is a normalized tight super frame wavelet of length  for  2 (R) ⊕ }.In case of the action of  1 over ,   , and SW NT  we obtain that the action is free but orbits are not isometric to  1 .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.

Prerequisites
For a generic countable (or finite) index set J such as N, Z, N ∪ N, and Z × Z, a collection of elements Φ = {  :  ∈ J} in a separable Hilbert space  is called a frame if there exist constants  and , 0 <  ≤  < ∞, such that The optimal constants (maximal for  and minimal for ) ( This shows that the elements of a frame need not be normal but they must have an upper bound. Definition 1 (see [4]).Let  be an × expansive matrix such that Z  ⊂ Z  .Then a finite set . ., } is an orthonormal basis for  2 (R  ), where for  ∈  2 (R  ) one uses the convention If a multiwavelet Ψ 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 1.
The following result establishes a characterization of normalized tight frame multiwavelet.
The Fourier transform of  ∈  1 (R  ) ∩  2 (R  ) is defined by where ⟨, ⟩ denotes the real inner product.
One of the methods of constructing orthonormal wavelets is based on multiresolution analysis which is a family of closed subspaces of  2 (R  ) satisfying certain properties.
We have the following result analogous to that as in the case of one dimension.
The following theorem characterizes all MSF multiwavelets.

, 𝐿 is an orthonormal multiwavelet with the dilation matrix 𝐴 if and only if
In [8], 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  1 ,  2 , . . .,   are normalized tight frame wavelets for  2 (R).One will call the -tuple ( 1 , . . .,   ) a super-wavelet of length  if Han and Larson in their memoirs [8] proved that, for each  ( can be ∞), there is a super-wavelet of length .
From Brouwer's fixed point theorem, it follows that Fix for 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 [10].This is what led to the notion of the complete invariance property.Formally, we have the following.
Definition 10 (see [1]).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 [2]).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 [2]).A space  has the CIPH if it satisfies the following conditions: (i)  1 acts on  freely; (ii)  possesses a bounded metric such that each orbit is isometric to the unit circle.

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  1 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  1 acts freely on a metric space  and orbits are isometric to circles of finite radii, then  has the CIPH.

Theorem 13. A space 𝑋 has the CIPH if it satisfies the following conditions:
(i)  1 acts on  freely; (ii)  possesses a bounded metric such that each orbit is isometric to  1  , a circle of radius , where 0 <  ≤  for some  > 0.
Thus  *  () =  *  () and hence, for some integer , By the triangle inequality applying over , ,  we have and so Thus the equation  + () − () = 2 holds only for  = 0 and hence  = 0. Since  is an orbit wise one-one map and a homeomorphism of  1   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 [2].
Theorem 14.The space   has the CIPH.
For the continuity of  at (  , Ψ), we simply observe that where The orbit of Ψ is given by which is isometric to  1  , the circle of radius 0 <  ≤ √ , via the map which sends   ⋅ Ψ to   , where  = ‖Ψ‖.Thus from Theorem 13 it follows that the space  NT has the CIPH.

Corollary 15. If 𝐴 is an expansive matrix and 𝐿 ≥ 1 is an integer, then the space
has the CIPH.
Proof.Note that  ⊂  NT .The restriction / of the action  to  is a free action.The orbit of Ψ is isometric to  1 √ , the circle of radius √ , via the map sending   ⋅Ψ to √   , where √  = ‖Ψ‖.Thus from Theorem 13 it follows that the space  has the CIPH.
In the case of tight frame, the frame bounds A and B are equal but need not be 1.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   0 has the CIPH.

Super-Wavelets and the CIPH
The concept of super-wavelets was first introduced and studied in [8].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.
The orbit of  is given by which is isometric to  1 , via the map where (  1 ,  1 ) ∈  1 × SW NT  and ‖‖ ≤ 1.The orbit of  is isometric to  1  , 0 <  ≤ 1, where  = ‖‖.Thus from Theorem 13 it follows that the space SW NT  has the CIPH.
Hilbert space is a normalized tight frame.Notice that, for a nonzero element   of a frame Φ in H, the following inequality holds: are called the frame bounds. is called a lower frame bound and  is called an upper frame bound of the frame.The frame {  :  ∈ J} is called a tight frame if  =  and is called normalized tight frame if  =  = 1.Any orthonormal basis in a ), where  and  are frame bounds of the frame generated by Ψ.Now, we show that   ⋅ Ψ is an element of   0 .That is,  :  1 ×   0 →   0 (30) defined by (  , Ψ) =   ⋅ Ψ is well defined and describes a free action of  1 on   0 .The continuity of  at (  , Ψ) follows by noting that       (  , Ψ) −  (  1 , Ψ 1 )  1 , Ψ 1 ) ∈  1 ×   0 and ‖Ψ‖ = √      1 2(R  ).