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≤LL2(Rn), containing all orthonormal multiwavelets on Rn inL-tuple form, WT⊂∏1≤j≤LL2(Rn), containing all tight frame multiwavelets on Rn in L-tuple form, SWn={(η1,…,ηn):(η1,…,ηn)is a super-wavelet of lengthn for L2(R)⊕n}, and SWnNT={(η1,…,ηn): (η1,…,ηn)is a normalized tight super frame wavelet of length nforL2(R)⊕n}. In case of the action of S1 over W, WT, and SWnNT 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={ϕj:j∈J} in a separable Hilbert spaceHis called a frame if there exist constants A and B, 0<A≤B<∞, such that
(1)A∥f∥2≤∑j∈J|〈f,ϕj〉|2≤B∥f∥2,∀f∈H.
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 {ϕj: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 ϕk of a frame Φ in H, the following inequality holds:
(2)∥ϕk∥≤B,∀k∈J.
This follows by noting that
(3)∥ϕk∥4=|〈ϕk,ϕk〉|2≤∑j∈J|〈ϕk,ϕj〉|2≤B∥ϕk∥2.
This shows that the elements of a frame need not be normal but they must have an upper bound.

Definition 1 (see [<xref ref-type="bibr" rid="B13">4</xref>]).

LetAbe an n×n expansive matrix such that AZn⊂Zn. Then a finite set Ψ={ψ1,…,ψL}⊂L2(Rn) is called an orthonormal multiwavelet if the collection A(Ψ)={ψj,kl:j∈Z,k∈Zn,l=1,…,L} is an orthonormal basis for L2(Rn), where for ψ∈L2(Rn) one uses the convention
(4)ψj,k=|detA|j/2ψ(Aj·-k).

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)={ψj,kl:j∈Z,k∈Zn,l=1,…,L} is a normalized tight frame, then the set Ψ={ψ1,…,ψL}⊂L2(Rn) 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 [<xref ref-type="bibr" rid="B2">5</xref>]).

Suppose Ψ={ψ1,…,ψL}⊂L2(Rn). Then the collection A(Ψ)={ψj,kl:j∈Z,k∈Zn,l=1,…,L} with a dilationAis a normalized tight frame if and only if

∑l=1L∑j∈Z|ψ^l(Bjξ)|2=1, for a.e.ξ∈Rn, where B is the transpose of A,

tq(ξ)≡∑l=1L∑j≥0ψ^l(Bjξ)ψ^l(Bj(ξ+q))¯=0, for q∈Zn∖BZn and a.e.ξ∈Rn.

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

The Fourier transform off∈L1(Rn)∩L2(Rn)is defined by
(5)f^(ξ)=∫Rnf(x)e-2πi〈ξ,x〉dx,ξ∈Rn,
where 〈ξ,x〉 denotes the real inner product.

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

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

Let Ψ={ψ1,…,ψL}⊂L2(Rn) be an orthonormal multiwavelet associated with a dilationA.

Then
(6)DΨ(ξ)=∑l=1L∑j∈N∑k∈Zn|ψ^l(Bj(ξ+k))|2,fora.e.ξ∈Rn,
describes the dimension functionDΨforΨ, whereBis the transpose ofA.

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

Theorem 3 (see [<xref ref-type="bibr" rid="B1">6</xref>]).

An orthonormal multiwaveletΨ∈L2(Rn)is an MRA multiwavelet if and only ifDΨ(ξ)=1, for a.e.ξ∈Rn.

Definition 4 (see [<xref ref-type="bibr" rid="B1">6</xref>]).

An MSF (minimally supported frequency) multiwavelet (of order L) is an orthonormal multiwavelet Ψ={ψ1,…,ψL} such that |ψ^l|=χWl for some measurable set Wl⊂Rn,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 [<xref ref-type="bibr" rid="B5">7</xref>]).

A set Ψ={ψ1,…,ψL}⊂L2(Rn) such that |ψ^l|=χWl for l=1,…,L is an orthonormal multiwavelet with the dilation matrixAif and only if
(7)∑k∈ZnχWl(ξ+k)χWl′(ξ+k)=δl,l′a.e.ξ∈Rn,l,l′=i,…,L,∑l=1L∑j∈ZχWl(Bjξ)=1,a.e.,ξ∈Rn,B=AT.

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,…,ηn are normalized tight frame wavelets for L2(R). One will call then-tuple (η1,…,ηn) a super-wavelet of lengthnif
(8){DjTkη1⊕⋯⊕DjTkηn:j,k∈Z}
is an orthonormal basis for L2(R)⊕⋯⊕L2(R) (say,L2(R)⊕n), where
(9)Djf(x)=2j/2f(2jx),Tkf(x)=f(x-k)forf∈L2(R).

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,…,ηn∈L2(R). Then (η1,…,ηn) is a super-wavelet of lengthnif and only if the following equations hold:

∑j∈Z|η^i(2jξ)|2=1,for a.e.ξ∈R,i=1,…,n,

∑j=0∞η^i(2jξ)η^i(2j(ξ+k))¯=0, for a.e. ξ∈R,k∈2Z+1,i=1,…,n,

∑j∈Z∑i=1n|η^i(ξ+k)|2=1, for a.e. ξ∈R,

∑j=0∞∑i=1mη^i(2j(ξ+k))η^i(ξ+k)¯=0, for a.e. ξ∈R,j∈N.

Definition 8.

A super-wavelet(η1,…,ηn)is said to be an MRA super-wavelet if every ηi(i=1,…,n) is an MRA frame wavelet.

Definition 9 (see [<xref ref-type="bibr" rid="B15">9</xref>]).

Suppose that (η1,η2,…,ηn)∈L2(R)⊕n. One will call then-tuple (η1,…,ηn) a normalized tight super frame wavelet of length n if
(10){DjTkη1⊕⋯⊕DjTkηn:j,k∈Z}
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 [<xref ref-type="bibr" rid="B12">1</xref>]).

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 [<xref ref-type="bibr" rid="B9">2</xref>]).

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 [<xref ref-type="bibr" rid="B9">2</xref>]).

A space X has the CIPH if it satisfies the following conditions:

S1 acts on X freely;

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:

S1 acts on X freely;

X possesses a bounded metric such that each orbit is isometric to Sr1, 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
(11)f(x)=x*eia(x),x∈X.

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 Sr1,0<r≤L, and, for some real number t,y=z*eit with d(y,z)=|rt|≤πL.

Thus y*eia(y)=z*eia(z) and hence, for some integer n,
(12)t+a(y)-a(z)=2πn.
By the triangle inequality applying over y, z, A we have
(13)2L|a(y)-a(z)|≤|rt|≤πL,
and so
(14)|a(y)-a(z)|≤π2.
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 Sr1 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
(15)WNT={(ψ1,…,ψL)∈∏1≤j≤LL2(Rn):{ψ1,…,ψL}mmk∏1≤j≤Lis a normalized tight frame multiwavelet}
is a metric space with the natural metric d defined by
(16)d2(Ψ,Φ)=∑l=1L∥ψl-ϕl∥22,
where Ψ=(ψ1,…,ψL) and Φ=(ϕ1,…,ϕL).

Theorem 14.

The space WNT has the CIPH.

Proof.

The characterization of a normalized tight frame multiwavelet (Theorem 2) provides that ifeiθ∈S1 and Ψ=(ψ1,…,ψL)∈WNT, then, eiθ·Ψ=(eiθ·ψ1,…,eiθ·ψL)∈WNT. For Ψ∈WNT the collection A(Ψ)={ψj,kl:j∈Z,k∈Zn,l=1,…,L} forms a normalized tight frame for L2(Rn). This shows that Ψ cannot be a zero function in L2(Rn). Thus, the function η:S1×WNT→WNT defined by
(17)η(eiθ,Ψ)=eiθ·Ψ,eiθ∈S1,Ψ∈WNT
is a free action of S1 on WNT.

For the continuity of η at (eiθ,Ψ), we simply observe that
(18)∥η(eiθ,Ψ)-η(eiθ1,Ψ1)∥≤|eiθ-eiθ1|∥Ψ∥+∥Ψ-Ψ1∥≤|eiθ-eiθ1|L+∥Ψ-Ψ1∥,
where (eiθ1,Ψ1)∈S1×WNT and
(19)∥Ψ∥≤∥ψ1∥22+⋯+∥ψL∥22≤L.
The orbit of Ψ is given by
(20)η(S1×{Ψ})={η(eiθ,Ψ):θ∈[0,2π)},
which is isometric to Sr1, the circle of radius 0<r≤L, via the map
(21)φ:η(S1×{Ψ})⟶Sr1
which sends eiθ·Ψ to reiθ, where r=∥Ψ∥. Thus from Theorem 13 it follows that the space WNT has the CIPH.

Corollary 15.

If A is an expansive matrix and L≥1 is an integer, then the space
(22)W={(ψ1,…,ψL)∈∏1≤j≤LL2(Rn):{ψ1,…,ψL}mmi∏1≤j≤Lisanorthonormalmultiwavelet}
has the CIPH.

Proof.

Note that W⊂WNT. The restriction η/W of the action η to W is a free action. The orbit of Ψ is isometric to SL1, the circle of radius L, via the map sending eiθ·Ψ to Leiθ, where L=∥Ψ∥. Thus from Theorem 13 it follows that the space W has the CIPH.

Remark 16.

Let WM={(ψ1,…,ψL)∈W:{ψ1,…,ψL}is an MRA multiwavelet} and WS={(ψ1,…,ψL)∈W:{ψ1,…,ψL} is an MSF multiwavelet}. By noting that the dimension function ofΨ,D(Ψ), is equal to D(η(eiθ,Ψ)), it follows from Theorem 3 that Ψ is an MRA wavelet if and only if η(eiθ,Ψ) is an MRA wavelet. Also, we note that Ψ is an MSF wavelet if and only if η(eiθ,Ψ) is an MSF wavelet. Thus WM, WS, and WM∩WS are invariant sets in W with respect to the action of topological group S1. The orbits of these invariant sets remain isometric to SL1. Thus from Theorem 13 it follows that the spaces WM, WS, and WM∩WS 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
(23)WT={(ψ1,…,ψL)∈∏1≤j≤LL2(Rn):{ψ1,…,ψL}mmi∏1≤j≤Lis a tight frame multiwavelet},
then we have the following result.

Theorem 17.

The space WT has the CIPH.

Theorem 18.

Let
(24)WB0={(ψ1,…,ψL)∈∏1≤j≤LL2(Rn):{ψ1,…,ψL}mmiisaframemultiwaveletwiththemmi∏1≤j≤LupperframeboundboundedbyB0}.
Then the space WB0 has the CIPH.

Proof.

Let Ψ=(ψ1,…,ψL)∈WB0; that is, the collection
(25)A(Ψ)={Ψj,kl:j∈Z,k∈Znandl=1,…,L}
is a frame of L2(Rn).

Then
(26)A∥f∥22≤∑l=1L∑j∈Z∑k∈Zn|〈f,ψj,kl〉|2≤B∥f∥22,
for all f∈L2(Rn), where A and B are frame bounds of the frame generated by Ψ.

Now, we show that eiθ·Ψ is an element of WB0. That is,
(27)A∥f∥22≤∑l=1L∑j∈Z∑k∈Zn|〈f,eiθψj,kl〉|2≤B∥f∥22,
for all f∈L2(Rn).

Note that
(28)〈f,eiθψj,kl〉=∫feiθψj,kl¯=eiθ¯∫fψj,kl¯.
Hence we have
(29)|〈f,ψj,kl〉|2=|〈f,eiθψj,kl〉|2.
Thus the map
(30)η:S1×WB0⟶WB0
defined by η(eiθ,Ψ)=eiθ·Ψ is well defined and describes a free action of S1 on WB0.

The continuity of η at (eiθ,Ψ) follows by noting that
(31)∥η(eiθ,Ψ)-η(eiθ1,Ψ1)∥≤|eiθ-eiθ1|LB0+∥Ψ-Ψ1∥,
where (eiθ1,Ψ1)∈S1×WB0 and
(32)∥Ψ∥=∥ψ1∥22+⋯+∥ψL∥22≤LB0.

The orbit of Ψ is isometric to Sr1, the circle of radius 0<r≤LB0, via the map sending eiθ·Ψ to reiθ, 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 SWn defined by
(33)SWn={L2(R)⊕nϑ=(η1,…,ηn):(η1,…,ηn)mmisasuper-waveletoflengthnforL2(R)⊕n}.
Then the space SWn has the CIPH.

Proof.

Let ϑ=(η1,…,ηn) be an element of SWn. From Theorem 7 it follows that eiθ·ϑ=(eiθη1,…,eiθηn) remains in SWn, where eiθ∈S1. Thus the map
(34)η:S1×SWn⟶SWn
defined by
(35)η(eiθ,ϑ)=eiθ·ϑ,ϑ∈SWn,eiθ∈S1
is a free action.

The continuity of η at (eiθ,ϑ) follows by noting that
(36)∥η(eiθ,ϑ)-η(eiθ1,ϑ1)∥≤|eiθ-eiθ1|+∥ϑ-ϑ1∥,
where (eiθ1,ϑ1)∈S1×SWn and ∥ϑ∥=1.

The orbit of ϑ is given by
(37)η(S1×{ϑ})={η(eiθ,ϑ):θ∈[0,2π)},
which is isometric to S1, via the map
(38)φ:η(S1×{ϑ})⟶S1,
which sends eiθ·ϑ to eiθ. Thus from Theorem 12 it follows that the space SWn has the CIPH.

Remark 20.

If SWnM={(η1,…,ηn):(η1,…,ηn) is an MRA super-wavelet of length n forL2(R)⊕n}, then, for each (η1,…,ηn)∈SWnM, (eiθη1,…,eiθηn) is also an MRA super-wavelet. Thus SWnMis 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 SWnM has the CIPH.

Theorem 21.

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

Proof.

Letϑ=(η1,…,ηn)∈SWnNT. Forf=(f1,…,fn)∈L2(R)⊕nwe have
(39)∥f∥2=∑j∈Z∑k∈Z|〈f,DjTkη1⊕⋯⊕DjTkηn〉|2=∑j∈Z∑k∈Z|〈f1,DjTkη1〉+⋯+〈fn,DjTkηn〉|2=∑j∈Z∑k∈Z|〈f1,DjTkeiθη1〉+⋯+〈fn,DjTkeiθηn〉|2.
This shows that eiθ·ϑ=(eiθη1,…,eiθηn) remains in SWnNT, where eiθ∈S1.

Thus the map
(40)η:S1×SWnNT⟶SWnNT
defined by
(41)η(eiθ,ϑ)=eiθ·ϑ,ϑ∈SWnNT,eiθ∈S1
is a free action.

The continuity of η at (eiθ,ϑ) follows by noting that
(42)∥η(eiθ,ϑ)-η(eiθ1,ϑ1)∥=∥eiθ·ϑ-eiθ1·ϑ1∥≤|eiθ-eiθ1|+∥ϑ-ϑ1∥,
where (eiθ1,ϑ1)∈S1×SWnNT and ∥ϑ∥≤1.

The orbit ofϑis isometric to Sr1,0<r≤1, where r=∥ϑ∥. Thus from Theorem 13 it follows that the space SWnNT has the CIPH.

For a d×d, real expansive matrix A, let DA and Tk(k∈Rd) be the unitary operators onL2(Rd)defined by
(43)(DAf)(x)=|A|1/2f(Ax),miTkf(x)=f(x-k),forf∈L2(Rd).
Then we have the following.

Definition 22 (see [<xref ref-type="bibr" rid="B15">9</xref>]).

Suppose that η1,…,ηn areA-dilation single normalized tight frame wavelets. One calls the n-tuple (η1,…,ηn) an A-dilation normalized tight super frame wavelet of lengthnif
(44){DAjTkη1⊕⋯⊕DAjTkηn:j∈Z,k∈Zd}
is anA-dilation normalized tight frame for L2(Rd)⊕n.

Thus from the above definition analogous result to Theorem 21 holds in case of higher dimension as well.

Remark 23.

Consider the set
(45)S={W,WM,WS,WM∩WS,mmWNT,WT,WB0,SWn,SWnM,SWnNT}.
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.

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