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 lengthn, and all normalized tight super frame wavelets of lengthn.
Highlights
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]
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 S1 on 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.”
Summary
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. 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 S1 on W and obtained each orbit isometric to S1. We have proved that the result of Martin stated above is true for orbits isometric to a circle of finite radius
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