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

Abstract

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.

1. Introduction

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 circle𝑆¹on 𝒲 and obtained each orbit isometric to 𝑆¹. 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 𝑋 has the CIPH if it satisfies the following conditions: (i) 𝑆¹ acts on 𝑋 freely. (ii) 𝑋 possesses a bounded metric such that each orbit is isometric to 𝑆¹.”

In this paper, we study the complete invariance property with respect to homeomorphism over the spaces 𝑊⊂∏_{1≤𝑗≤𝐿}𝐿²(ℝ^𝑛), containing all orthonormal multiwavelets on ℝ^𝑛 in𝐿-tuple form, 𝑊_𝑇⊂∏_{1≤𝑗≤𝐿}𝐿²(ℝ^𝑛), containing all tight frame multiwavelets on ℝ^𝑛 in 𝐿-tuple form, 𝒮𝒲_𝑛={(𝜂₁,…,𝜂_𝑛)∶(𝜂₁,…,𝜂_𝑛)is a super-wavelet of length𝑛 for 𝐿²(ℝ)^⊕𝑛}, and 𝒮𝒲^NT_𝑛={(𝜂₁,…,𝜂_𝑛): (𝜂₁,…,𝜂_𝑛)is a normalized tight super frame wavelet of length 𝑛for𝐿²(ℝ)^⊕𝑛}. In case of the action of 𝑆¹ over 𝑊, 𝑊_𝑇, and 𝒮𝒲^NT_𝑛 we obtain that the action is free but orbits are not isometric to 𝑆¹. 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.

2. Prerequisites

For a generic countable (or finite) index set 𝕁 such as ℕ, ℤ, ℕ∪ℕ, and ℤ×ℤ, a collection of elements Φ={𝜙_𝑗∶𝑗∈𝕁} in a separable Hilbert space𝐻is called a frame if there exist constants 𝐴 and 𝐵, 0<𝐴≤𝐵<∞, such that 𝐴𝑓²≤_𝑗∈𝕁𝑓,𝜙_𝑗²≤𝐵𝑓²,∀𝑓∈𝐻. (1) The optimal constants (maximal for𝐴and minimal for 𝐵) are called the frame bounds. 𝐴 is called a lower frame bound and 𝐵 is called an upper frame bound of the frame. The frame {𝜙_𝑗∶𝑗∈𝕁} is called a tight frame if 𝐴=𝐵 and is called normalized tight frame if 𝐴=𝐵=1. Any orthonormal basis in a Hilbert space is a normalized tight frame. Notice that, for a nonzero element 𝜙_𝑘 of a frame Φ in ℍ, the following inequality holds: 𝜙_𝑘≤√𝐵,∀𝑘∈𝕁. (2) This follows by noting that 𝜙_𝑘⁴=𝜙_𝑘,𝜙_𝑘²≤_𝑗∈𝕁𝜙_𝑘,𝜙_𝑗²≤𝐵𝜙_𝑘². (3) This shows that the elements of a frame need not be normal but they must have an upper bound.

Definition 1 (see [4]). Let𝐴be an 𝑛×𝑛 expansive matrix such that 𝐴ℤ^𝑛⊂ℤ^𝑛. Then a finite set Ψ={𝜓¹,…,𝜓^𝐿}⊂𝐿²(ℝ^𝑛) is called an orthonormal multiwavelet if the collection 𝒜(Ψ)={𝜓^𝑙_𝑗,𝑘∶𝑗∈ℤ,𝑘∈ℤ^𝑛,𝑙=1,…,𝐿} is an orthonormal basis for 𝐿²(ℝ^𝑛), where for 𝜓∈𝐿²(ℝ^𝑛) one uses the convention 𝜓_𝑗,𝑘=|det𝐴|^𝑗/2𝜓𝐴^𝑗⋅−𝑘. (4)
If a multiwavelet Ψ consists of a single element 𝜓, then we say that 𝜓 is a wavelet. By an expansive matrix 𝐴, we mean a square matrix the moduli of whose eigenvalues are greater than1.
If the collection 𝒜(Ψ)={𝜓^𝑙_𝑗,𝑘∶𝑗∈ℤ,𝑘∈ℤ^𝑛,𝑙=1,…,𝐿} is a normalized tight frame, then the set Ψ={𝜓¹,…,𝜓^𝐿}⊂𝐿²(ℝ^𝑛) is called a normalized tight frame multiwavelet. Similarly, Ψ is called a tight frame multiwavelet when the above collection 𝒜(Ψ) is a tight frame and a frame multiwavelet when the above collection 𝒜(Ψ) is a frame.
The following result establishes a characterization of normalized tight frame multiwavelet.

Theorem 2 (see [5]). Suppose Ψ={𝜓¹,…,𝜓^𝐿}⊂𝐿²(ℝ^𝑛). Then the collection 𝒜(Ψ)={𝜓^𝑙_𝑗,𝑘∶𝑗∈ℤ,𝑘∈ℤ^𝑛,𝑙=1,…,𝐿} with a dilation𝐴is a normalized tight frame if and only if(i)∑^𝐿_𝑙=1∑_𝑗∈ℤ|𝜓^𝑙(𝐵^𝑗𝜉)|²=1, for 𝑎.𝑒.𝜉∈ℝ^𝑛, where 𝐵 is the transpose of 𝐴,(ii)𝑡_𝑞(𝜉)≡∑^𝐿_𝑙=1∑_𝑗≥0𝜓^𝑙(𝐵^𝑗𝜉)𝜓^𝑙(𝐵^𝑗(𝜉+𝑞))=0, for 𝑞∈ℤ^𝑛⧵𝐵ℤ^𝑛 and a.e.𝜉∈ℝ^𝑛.

In particularΨis a multiwavelet if and only if the above conditions hold and ‖𝜓^𝑙‖=1 for all 𝑙=1,…,𝐿.

The Fourier transform of𝑓∈𝐿¹(ℝ^𝑛)∩𝐿²(ℝ^𝑛)is defined by 𝑓(𝜉)=_ℝ𝑛𝑓(𝑥)𝑒^{−2𝜋𝑖⟨𝜉,𝑥⟩}𝑑𝑥,𝜉∈ℝ^𝑛, (5) where ⟨𝜉,𝑥⟩ denotes the real inner product.

Since𝐿¹(ℝ^𝑛)∩𝐿²(ℝ^𝑛)is a dense subset of𝐿²(ℝ^𝑛), this definition extends uniquely to𝐿²(ℝ^𝑛).

One of the methods of constructing orthonormal wavelets is based on multiresolution analysis which is a family of closed subspaces of 𝐿²(ℝ^𝑛) satisfying certain properties.

Let Ψ={𝜓¹,…,𝜓^𝐿}⊂𝐿²(ℝ^𝑛) be an orthonormal multiwavelet associated with a dilation𝐴.

Then 𝐷_Ψ(𝜉)=^𝐿_{𝑙=1𝑗∈ℕ𝑘∈ℤ𝑛}𝜓^𝑙𝐵^𝑗(𝜉+𝑘)²,fora.e.𝜉∈ℝ^𝑛, (6) describes the dimension function𝐷_ΨforΨ, where𝐵is the transpose of𝐴.

We have the following result analogous to that as in the case of one dimension.

Theorem 3 (see [6]). An orthonormal multiwaveletΨ∈𝐿²(ℝ^𝑛)is an MRA multiwavelet if and only if𝐷_Ψ(𝜉)=1, for a.e.𝜉∈ℝ^𝑛.

Definition 4 (see [6]). An MSF (minimally supported frequency) multiwavelet (of order L) is an orthonormal multiwavelet Ψ={𝜓¹,…,𝜓^𝐿} such that |𝜓^𝑙|=𝜒_𝑊𝑙 for some measurable set 𝑊_𝑙⊂ℝ^𝑛,𝑙=1,…,𝐿. An MSF multiwavelet of order 1 is simply referred to as an MSF wavelet.
The following theorem characterizes all MSF multiwavelets.

Theorem 5 (see [7]). A set Ψ={𝜓¹,…,𝜓^𝐿}⊂𝐿²(ℝ^𝑛) such that |𝜓^𝑙|=𝜒_𝑊𝑙 for 𝑙=1,…,𝐿 is an orthonormal multiwavelet with the dilation matrix𝐴if and only if _{𝑘∈ℤ𝑛}𝜒_𝑊𝑙(𝜉+𝑘)𝜒_𝑊𝑙′(𝜉+𝑘)=𝛿_𝑙,𝑙′𝑎.𝑒.𝜉∈ℝ^𝑛,𝑙,𝑙^′=𝑖,…,𝐿,_{𝐿𝑙=1𝑗∈ℤ}𝜒_𝑊𝑙𝐵^𝑗𝜉=1,𝑎.𝑒.,𝜉∈ℝ^𝑛,𝐵=𝐴^𝑇. (7)

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 𝜂₁,𝜂₂,…,𝜂_𝑛 are normalized tight frame wavelets for 𝐿²(ℝ). One will call the𝑛-tuple (𝜂₁,…,𝜂_𝑛) a super-wavelet of length𝑛if 𝐷^𝑗𝑇^𝑘𝜂₁⊕⋯⊕𝐷^𝑗𝑇^𝑘𝜂_𝑛∶𝑗,𝑘∈ℤ (8) is an orthonormal basis for 𝐿²(ℝ)⊕⋯⊕𝐿²(ℝ) (say,𝐿²(ℝ)^⊕𝑛), where 𝐷^𝑗𝑓(𝑥)=2^𝑗/2𝑓2^𝑗𝑥,𝑇^𝑘𝑓(𝑥)=𝑓(𝑥−𝑘)for𝑓∈𝐿²(ℝ). (9)
Han and Larson in their memoirs [8] proved that, for each𝑛(𝑛can be∞), there is a super-wavelet of length𝑛.
Following is a characterization of a super-wavelet of length𝑛.

Theorem 7. Let 𝜂₁,…,𝜂_𝑛∈𝐿²(ℝ). Then (𝜂₁,…,𝜂_𝑛) is a super-wavelet of length𝑛if and only if the following equations hold: (i)∑_𝑗∈ℤ|𝜂_𝑖(2^𝑗𝜉)|²=1,for a.e.𝜉∈ℝ,𝑖=1,…,𝑛,(ii)∑^∞_𝑗=0𝜂_𝑖(2^𝑗𝜉)𝜂_𝑖(2^𝑗(𝜉+𝑘))=0, for a.e. 𝜉∈ℝ,𝑘∈2ℤ+1,𝑖=1,…,𝑛,(iii)∑_𝑗∈ℤ∑^𝑛_𝑖=1|𝜂_𝑖(𝜉+𝑘)|²=1, for a.e. 𝜉∈ℝ,(iv)∑^∞_𝑗=0∑^𝑚_𝑖=1𝜂_𝑖(2^𝑗(𝜉+𝑘))𝜂_𝑖(𝜉+𝑘)=0, for a.e. 𝜉∈ℝ,𝑗∈ℕ.

Definition 8. A super-wavelet(𝜂₁,…,𝜂_𝑛)is said to be an MRA super-wavelet if every 𝜂_𝑖(𝑖=1,…,𝑛) is an MRA frame wavelet.

Definition 9 (see [9]). Suppose that (𝜂₁,𝜂₂,…,𝜂_𝑛)∈𝐿²(ℝ)^⊕𝑛. One will call the𝑛-tuple (𝜂₁,…,𝜂_𝑛) a normalized tight super frame wavelet of length 𝑛 if 𝐷^𝑗𝑇^𝑘𝜂₁⊕⋯⊕𝐷^𝑗𝑇^𝑘𝜂_𝑛∶𝑗,𝑘∈ℤ (10) is a normalized tight frame for 𝐿²(ℝ)^⊕𝑛.
For a self-continuous map 𝑓 on a topological space 𝑋, Fix𝑓 denotes the set of all fixed points of 𝑓. A point 𝑥∈𝑋 is called a fixed point of 𝑓 if 𝑓(𝑥)=𝑥. In case 𝑋 is Hausdorff, Fix𝑓 is a closed set.
From Brouwer’s fixed point theorem, it follows that Fix𝑓for a self-continuous map 𝑓 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 𝑋 is said to possess the complete invariance property (CIP) if each of its nonempty closed sets is Fix𝑓, for some self-continuous map 𝑓 on  𝑋.

Definition 11 (see [2]). A topological space 𝑋 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, Fix𝑓, for some self-homeomorphism 𝑓 on 𝑋.

Theorem 12 (see [2]). A space 𝑋 has the CIPH if it satisfies the following conditions: (i)𝑆¹ acts on 𝑋 freely;(ii)𝑋 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 𝑋 we need a free 𝑆¹ action on 𝑋 having orbits isometric to the unit circle. This result does not provide any information about the CIPH over 𝑋 in case the radii of orbits are different from unity.

Below we modify the above result and show that if 𝑆¹ acts freely on a metric space 𝑋 and orbits are isometric to circles of finite radii, then 𝑋 has the CIPH.

Theorem 13. A space 𝑋 has the CIPH if it satisfies the following conditions:(i)𝑆¹ acts on 𝑋 freely;(ii)𝑋 possesses a bounded metric such that each orbit is isometric to 𝑆¹_𝑟, a circle of radius 𝑟, where 0<𝑟≤𝐿 for some 𝐿>0.

Proof. Let (𝑋,𝑑) be a metric space with 𝑑≤3𝜋𝐿 and let ∗∶𝑋×𝑆¹→𝑋 be the action satisfying conditions (i) and (ii). For a nonempty closed set 𝐴 in 𝑋 set 𝑎(𝑥)=(1/2𝐿)𝑑(𝑥,𝐴) and define 𝑓∶𝑋→𝑋 by 𝑓(𝑥)=𝑥∗𝑒^{𝑖𝑎(𝑥)},𝑥∈𝑋. (11)
Since 0<𝑎(𝑥)<2𝜋 if 𝑥∉𝐴, it follows that Fix𝑓=𝐴. To see that 𝑓 is one-one suppose 𝑓(𝑦)=𝑓(𝑧). Then𝑦and𝑧must lie on the same orbit isometric to the circle 𝑆¹_𝑟,0<𝑟≤𝐿, and, for some real number 𝑡,𝑦=𝑧∗𝑒^𝑖𝑡 with 𝑑(𝑦,𝑧)=|𝑟𝑡|≤𝜋𝐿.
Thus 𝑦∗𝑒^{𝑖𝑎(𝑦)}=𝑧∗𝑒^{𝑖𝑎(𝑧)} and hence, for some integer 𝑛, 𝑡+𝑎𝑦−𝑎(𝑧)=2𝜋𝑛. (12) By the triangle inequality applying over 𝑦, 𝑧, 𝐴 we have 2𝐿𝑎𝑦−𝑎(𝑧)≤|𝑟𝑡|≤𝜋𝐿, (13) and so 𝑎𝑦(𝑧≤𝜋−𝑎)2. (14) Thus the equation 𝑡+𝑎(𝑦)−𝑎(𝑧)=2𝜋𝑛 holds only for 𝑛=0 and hence 𝑡=0.
Since 𝑓 is an orbit wise one-one map and a homeomorphism of 𝑆¹_𝑟 into itself must be onto, it follows that 𝑓 is onto. In order to conclude that 𝑓 is a homeomorphism it suffices to show that 𝑓 is a closed mapping. For the remaining portion see the proof of Theorem 2.2 [2].

Let 𝐴 be an expansive matrix and 𝐿≥1 is an integer. Then the space 𝑊_NT=𝜓¹,…,𝜓^𝐿∈_{1≤𝑗≤𝐿}𝐿²ℝ^𝑛∶𝜓¹,…,𝜓^𝐿isanormalizedtightframemultiwavelet (15) is a metric space with the natural metric 𝑑 defined by 𝑑²(Ψ,Φ)=^𝐿_𝑙=1𝜓^𝑙−𝜙^𝑙2₂, (16) where Ψ=(𝜓¹,…,𝜓^𝐿) and Φ=(𝜙¹,…,𝜙^𝐿).

Theorem 14. The space 𝑊_𝑁𝑇 has the CIPH.

Proof. The characterization of a normalized tight frame multiwavelet (Theorem 2) provides that if𝑒^𝑖𝜃∈𝑆¹ and Ψ=(𝜓¹,…,𝜓^𝐿)∈𝑊_NT, then, 𝑒^𝑖𝜃⋅Ψ=(𝑒^𝑖𝜃⋅𝜓¹,…,𝑒^𝑖𝜃⋅𝜓^𝐿)∈𝑊_NT. For Ψ∈𝑊_NT the collection 𝒜(Ψ)={𝜓^𝑙_𝑗,𝑘∶𝑗∈ℤ,𝑘∈ℤ^𝑛,𝑙=1,…,𝐿} forms a normalized tight frame for 𝐿²(ℝ^𝑛). This shows that Ψ cannot be a zero function in 𝐿²(ℝ^𝑛). Thus, the function 𝜂∶𝑆¹×𝑊_NT→𝑊_NT defined by 𝜂𝑒^𝑖𝜃,Ψ=𝑒^𝑖𝜃⋅Ψ,𝑒^𝑖𝜃∈𝑆¹,Ψ∈𝑊_NT (17) is a free action of 𝑆¹ on 𝑊_NT.
For the continuity of 𝜂 at (𝑒^𝑖𝜃,Ψ), we simply observe that 𝜂𝑒^𝑖𝜃,Ψ−𝜂𝑒^𝑖𝜃1,Ψ₁≤𝑒^𝑖𝜃−𝑒^𝑖𝜃1‖Ψ‖+Ψ−Ψ₁≤𝑒^𝑖𝜃−𝑒^𝑖𝜃1√𝐿+Ψ−Ψ₁, (18) where (𝑒^𝑖𝜃1,Ψ₁)∈𝑆¹×𝑊_NT and ‖Ψ‖≤𝜓¹²₂+⋯+𝜓^𝐿2₂≤√𝐿. (19) The orbit of Ψ is given by 𝜂𝑆¹×{Ψ}=𝜂𝑒^𝑖𝜃,Ψ∶𝜃∈[0,2𝜋), (20) which is isometric to 𝑆¹_𝑟, the circle of radius 0<𝑟≤√𝐿, via the map 𝜑∶𝜂𝑆¹×{Ψ}⟶𝑆¹_𝑟 (21) which sends 𝑒^𝑖𝜃⋅Ψ to 𝑟𝑒^𝑖𝜃, where 𝑟=‖Ψ‖. Thus from Theorem 13 it follows that the space 𝑊_NT has the CIPH.

Corollary 15. If 𝐴 is an expansive matrix and 𝐿≥1 is an integer, then the space 𝑊=𝜓¹,…,𝜓^𝐿∈_{1≤𝑗≤𝐿}𝐿²ℝ^𝑛∶𝜓¹,…,𝜓^𝐿𝑖𝑠𝑎𝑛𝑜𝑟𝑡ℎ𝑜𝑛𝑜𝑟𝑚𝑎𝑙𝑚𝑢𝑙𝑡𝑖𝑤𝑎𝑣𝑒𝑙𝑒𝑡 (22) has the CIPH.

Proof. Note that 𝑊⊂𝑊_NT. The restriction 𝜂/𝑊 of the action 𝜂 to 𝑊 is a free action. The orbit of Ψ is isometric to 𝑆¹_√𝐿, the circle of radius √𝐿, via the map sending 𝑒^𝑖𝜃⋅Ψ to √𝐿𝑒^𝑖𝜃, where √𝐿=‖Ψ‖. Thus from Theorem 13 it follows that the space 𝑊 has the CIPH.

Remark 16. Let 𝑊^𝑀={(𝜓¹,…,𝜓^𝐿)∈𝑊∶{𝜓¹,…,𝜓^𝐿}isanMRAmultiwavelet} and 𝑊^𝑆={(𝜓¹,…,𝜓^𝐿)∈𝑊∶{𝜓¹,…,𝜓^𝐿}isanMSFmultiwavelet}. By noting that the dimension function ofΨ,𝐷(Ψ), is equal to 𝐷(𝜂(𝑒^𝑖𝜃,Ψ)), it follows from Theorem 3 that Ψ is an MRA wavelet if and only if 𝜂(𝑒^𝑖𝜃,Ψ) is an MRA wavelet. Also, we note that Ψ is an MSF wavelet if and only if 𝜂(𝑒^𝑖𝜃,Ψ) is an MSF wavelet. Thus 𝑊^𝑀, 𝑊^𝑆, and 𝑊^𝑀∩𝑊^𝑆 are invariant sets in 𝑊 with respect to the action of topological group 𝑆¹. The orbits of these invariant sets remain isometric to 𝑆¹_√𝐿. Thus from Theorem 13 it follows that the spaces 𝑊^𝑀, 𝑊^𝑆, and 𝑊^𝑀∩𝑊^𝑆 have the CIPH.
In the case of tight frame, the frame bounds A and B are equal but need not be1. After a renormalization, we can assume A = B = 1. If we denote 𝑊_𝑇=𝜓¹,…,𝜓^𝐿∈_{1≤𝑗≤𝐿}𝐿²ℝ^𝑛∶𝜓¹,…,𝜓^𝐿isatightframemultiwavelet, (23) then we have the following result.

Theorem 17. The space 𝑊_𝑇 has the CIPH.

Theorem 18. Let 𝑊_𝐵0=𝜓¹,…,𝜓^𝐿∈_{1≤𝑗≤𝐿}𝐿²ℝ^𝑛∶𝜓¹,…,𝜓^𝐿𝑖𝑠𝑎𝑓𝑟𝑎𝑚𝑒𝑚𝑢𝑙𝑡𝑖𝑤𝑎𝑣𝑒𝑙𝑒𝑡𝑤𝑖𝑡ℎ𝑡ℎ𝑒𝑢𝑝𝑝𝑒𝑟𝑓𝑟𝑎𝑚𝑒𝑏𝑜𝑢𝑛𝑑𝑏𝑜𝑢𝑛𝑑𝑒𝑑𝑏𝑦𝐵₀. (24) Then the space 𝑊_𝐵0 has the CIPH.

Proof. Let Ψ=(𝜓¹,…,𝜓^𝐿)∈𝑊_𝐵0; that is, the collection 𝒜(Ψ)=Ψ^𝑙_𝑗,𝑘∶𝑗∈ℤ,𝑘∈ℤ^𝑛and𝑙=1,…,𝐿 (25) is a frame of 𝐿²(ℝ^𝑛).
Then 𝐴𝑓²₂≤^𝐿_{𝑙=1𝑗∈ℤ𝑘∈ℤ𝑛}𝑓,𝜓^𝑙_𝑗,𝑘²≤𝐵𝑓²₂, (26) for all 𝑓∈𝐿²(ℝ^𝑛), where 𝐴 and 𝐵 are frame bounds of the frame generated by Ψ.
Now, we show that 𝑒^𝑖𝜃⋅Ψ is an element of 𝑊_𝐵0. That is, 𝐴𝑓²₂≤^𝐿_{𝑙=1𝑗∈ℤ𝑘∈ℤ𝑛}𝑓,𝑒^𝑖𝜃𝜓^𝑙_𝑗,𝑘²≤𝐵𝑓²₂, (27) for all 𝑓∈𝐿²(ℝ^𝑛).
Note that 𝑓,𝑒^𝑖𝜃𝜓^𝑙_𝑗,𝑘=𝑓𝑒^𝑖𝜃𝜓^𝑙_𝑗,𝑘=𝑒^𝑖𝜃𝑓𝜓^𝑙_𝑗,𝑘. (28) Hence we have 𝑓,𝜓^𝑙_𝑗,𝑘²=𝑓,𝑒^𝑖𝜃𝜓^𝑙_𝑗,𝑘². (29) Thus the map 𝜂∶𝑆¹×𝑊_𝐵0⟶𝑊_𝐵0 (30) defined by 𝜂(𝑒^𝑖𝜃,Ψ)=𝑒^𝑖𝜃⋅Ψ is well defined and describes a free action of 𝑆¹ on 𝑊_𝐵0.
The continuity of 𝜂 at (𝑒^𝑖𝜃,Ψ) follows by noting that 𝜂𝑒^𝑖𝜃,Ψ−𝜂𝑒^𝑖𝜃1,Ψ₁≤𝑒^𝑖𝜃−𝑒^𝑖𝜃1𝐿𝐵₀+Ψ−Ψ₁, (31) where (𝑒^𝑖𝜃1,Ψ₁)∈𝑆¹×𝑊_𝐵0 and ‖Ψ‖=𝜓¹²₂+⋯+𝜓^𝐿2₂≤𝐿𝐵₀. (32)
The orbit of Ψ is isometric to 𝑆¹_𝑟, the circle of radius 0<𝑟≤𝐿𝐵₀, via the map sending 𝑒^𝑖𝜃⋅Ψ to 𝑟𝑒^𝑖𝜃, where 𝑟=‖Ψ‖; hence, the result is obtained.

4. 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.

Theorem 19. Let 𝑛≥2 be an integer. Consider the set 𝒮𝒲_𝑛 defined by 𝒮𝒲_𝑛=𝜗=𝜂₁,…,𝜂_𝑛∶𝜂₁,…,𝜂_𝑛𝑖𝑠𝑎𝑠𝑢𝑝𝑒𝑟-𝑤𝑎𝑣𝑒𝑙𝑒𝑡𝑜𝑓𝑙𝑒𝑛𝑔𝑡ℎ𝑛𝑓𝑜𝑟𝐿²(ℝ)^⊕𝑛. (33) Then the space 𝒮𝒲_𝑛 has the CIPH.

Proof. Let 𝜗=(𝜂₁,…,𝜂_𝑛) be an element of 𝒮𝒲_𝑛. From Theorem 7 it follows that 𝑒^𝑖𝜃⋅𝜗=(𝑒^𝑖𝜃𝜂₁,…,𝑒^𝑖𝜃𝜂_𝑛) remains in 𝒮𝒲_𝑛, where 𝑒^𝑖𝜃∈𝑆¹. Thus the map 𝜂∶𝑆¹×𝒮𝒲_𝑛⟶𝒮𝒲_𝑛 (34) defined by 𝜂𝑒^𝑖𝜃,𝜗=𝑒^𝑖𝜃⋅𝜗,𝜗∈𝒮𝒲_𝑛,𝑒^𝑖𝜃∈𝑆¹ (35) is a free action.
The continuity of 𝜂 at (𝑒^𝑖𝜃,𝜗) follows by noting that 𝜂𝑒^𝑖𝜃,𝜗−𝜂𝑒^𝑖𝜃1,𝜗₁≤𝑒^𝑖𝜃−𝑒^𝑖𝜃1+𝜗−𝜗₁, (36) where (𝑒^𝑖𝜃1,𝜗₁)∈𝑆¹×𝒮𝒲_𝑛 and ‖𝜗‖=1.
The orbit of 𝜗 is given by 𝜂𝑆¹×{𝜗}=𝜂𝑒^𝑖𝜃,𝜗∶𝜃∈[0,2𝜋), (37) which is isometric to 𝑆¹, via the map 𝜑∶𝜂𝑆¹×{𝜗}⟶𝑆¹, (38) which sends 𝑒^𝑖𝜃⋅𝜗 to 𝑒^𝑖𝜃. Thus from Theorem 12 it follows that the space 𝒮𝒲_𝑛 has the CIPH.

Remark 20. If 𝒮𝒲^𝑀_𝑛={(𝜂₁,…,𝜂_𝑛)∶(𝜂₁,…,𝜂_𝑛) is an MRA super-wavelet of length 𝑛 for𝐿²(ℝ)^⊕𝑛}, then, for each (𝜂₁,…,𝜂_𝑛)∈𝒮𝒲^𝑀_𝑛, (𝑒^𝑖𝜃𝜂₁,…,𝑒^𝑖𝜃𝜂_𝑛) is also an MRA super-wavelet. Thus 𝒮𝒲^𝑀_𝑛is an invariant set with respect to the action of 𝑆¹. Orbits of these invariant sets are isometric to the unit circle. Thus from Theorem 12 it follows that the space 𝒮𝒲^𝑀_𝑛 has the CIPH.

Theorem 21. Let 𝑛≥2 be an integer. Consider the set 𝒮𝒲_𝑛^𝑁𝑇 defined by 𝒮𝒲_𝑛^𝑁𝑇={𝜗=(𝜂₁,…,𝜂_𝑛): (𝜂₁,…,𝜂_𝑛)is a normalized tight super frame wavelet of length 𝑛 for 𝐿²(ℝ)^⊕𝑛}. Then the space 𝒮𝒲_𝑛^𝑁𝑇 has the CIPH.

Proof. Let𝜗=(𝜂₁,…,𝜂_𝑛)∈𝒮𝒲^NT_𝑛. For𝑓=(𝑓₁,…,𝑓_𝑛)∈𝐿²(ℝ)^⊕𝑛we have 𝑓²=_{𝑗∈ℤ𝑘∈ℤ}𝑓,𝐷^𝑗𝑇^𝑘𝜂₁⊕⋯⊕𝐷^𝑗𝑇^𝑘𝜂_𝑛²=_{𝑗∈ℤ𝑘∈ℤ}𝑓₁,𝐷^𝑗𝑇^𝑘𝜂₁+⋯+𝑓_𝑛,𝐷^𝑗𝑇^𝑘𝜂_𝑛²=_{𝑗∈ℤ𝑘∈ℤ}𝑓₁,𝐷^𝑗𝑇^𝑘𝑒^𝑖𝜃𝜂₁+⋯+𝑓_𝑛,𝐷^𝑗𝑇^𝑘𝑒^𝑖𝜃𝜂_𝑛². (39) This shows that 𝑒^𝑖𝜃⋅𝜗=(𝑒^𝑖𝜃𝜂₁,…,𝑒^𝑖𝜃𝜂_𝑛) remains in 𝒮𝒲^NT_𝑛, where 𝑒^𝑖𝜃∈𝑆¹.
Thus the map 𝜂∶𝑆¹×𝒮𝒲^NT_𝑛⟶𝒮𝒲^NT_𝑛 (40) defined by 𝜂𝑒^𝑖𝜃,𝜗=𝑒^𝑖𝜃⋅𝜗,𝜗∈𝒮𝒲^NT_𝑛,𝑒^𝑖𝜃∈𝑆¹ (41) is a free action.
The continuity of 𝜂 at (𝑒^𝑖𝜃,𝜗) follows by noting that 𝜂𝑒^𝑖𝜃,𝜗−𝜂𝑒^𝑖𝜃1,𝜗₁=𝑒^𝑖𝜃⋅𝜗−𝑒^𝑖𝜃1⋅𝜗₁≤𝑒^𝑖𝜃−𝑒^𝑖𝜃1+𝜗−𝜗₁, (42) where (𝑒^𝑖𝜃1,𝜗₁)∈𝑆¹×𝒮𝒲^NT_𝑛 and ‖𝜗‖≤1.
The orbit of𝜗is isometric to 𝑆¹_𝑟,0<𝑟≤1, where 𝑟=‖𝜗‖. Thus from Theorem 13 it follows that the space 𝒮𝒲^NT_𝑛 has the CIPH.

For a 𝑑×𝑑, real expansive matrix 𝐴, let 𝐷_𝐴 and 𝑇^𝑘 (𝑘∈ℝ^𝑑) be the unitary operators on𝐿²(ℝ^𝑑)defined by 𝐷_𝐴𝑓(𝑥)=|𝐴|^1/2𝑓(),𝑇𝐴𝑥_𝑘𝑓(𝑥)=𝑓(𝑥−𝑘),for𝑓∈𝐿²ℝ^𝑑. (43) Then we have the following.

Definition 22 (see [9]). Suppose that 𝜂₁,…,𝜂_𝑛 are𝐴-dilation single normalized tight frame wavelets. One calls the 𝑛-tuple (𝜂₁,…,𝜂_𝑛) an 𝐴-dilation normalized tight super frame wavelet of length𝑛if 𝐷^𝑗_𝐴𝑇^𝑘𝜂₁⊕⋯⊕𝐷^𝑗_𝐴𝑇^𝑘𝜂_𝑛∶𝑗∈ℤ,𝑘∈ℤ^𝑑 (44) is an𝐴-dilation normalized tight frame for 𝐿₂(ℝ^𝑑)^⊕𝑛.
Thus from the above definition analogous result to Theorem 21 holds in case of higher dimension as well.

Remark 23. Consider the set 𝒮=𝑊,𝑊^𝑀,𝑊^𝑆,𝑊^𝑀∩𝑊^𝑆,𝑊_NT,𝑊_𝑇,𝑊_𝐵0,𝒮𝒲_𝑛,𝒮𝒲^𝑀_𝑛,𝒮𝒲^NT_𝑛. (45) Then we have similar results to the one-dimensional orthonormal wavelet case provided in [3]. Let 𝑋,𝑌∈𝒮. Then the product space 𝑋×𝑌, the cylinder 𝑋×𝐼, where 𝐼 is the unit closed interval of the real line, the cone 𝐶(𝑋), and the suspension 𝑆(𝑋) have the CIP.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

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.