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. 1. 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 , of all one-dimensional orthonormal wavelets on and certain subsets of . They noticed a free action of the unit circle on 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 [2]: “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. 2. Prerequisites For a generic countable (or finite) index set such as , , , and , a collection of elements in a separable Hilbert space is called a frame if there exist constants and , , such that The optimal constants (maximal for and 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