In this article we establish theory of semi-orthogonal Parseval wavelets associated to generalized multiresolution analysis (GMRA) for the local field of positive characteristics (LFPC). By employing the properties of translation invariant spaces on the core space of GMRA we obtain a characterization of semi-orthogonal Parseval wavelets in terms of consistency equation for LFPC. As a consequence, we obtain a characterization of an orthonormal (multi)wavelet to be associated with an MRA in terms of multiplicity function as well as dimension function of a (multi)wavelet. Further, we provide characterizations of Parseval scaling functions, scaling sets and bandlimited wavelets together with a Shannon type multiwavelet for LFPC.

The concept of super‐wavelet was introduced by Balan, and Han and Larson over the field of real numbers which has many applications not only in engineering branches but also in different areas of mathematics. To develop this notion on local fields having positive characteristic we obtain characterizations of super‐wavelets of finite length as well as Parseval frame multiwavelet sets of finite order in this setup. Using the group theoretical approach based on coset representatives, further we establish Shannon type multiwavelet in this perspective while providing examples of Parseval frame (multi)wavelets and (Parseval frame) super‐wavelets. In addition, we obtain necessary conditions for decomposable and extendable Parseval frame wavelets associated to Parseval frame super‐wavelets.

Abstract. In this paper it is investigated as to when a nonempty invariant closed subset A of a S 1 -space X containing the set of stationary points (S) can be the fixed point set of an equivariant continuous selfmap on X and such space X is said to possess the Sequivariant complete invariance property (S-ECIP). It is also shown that if X is a metric space and S 1 acts on X × S 1 by the action (x, p) · q = (x, p · q), where p, q ∈ S 1 and x ∈ X, then the hyperspace 2 X×S 1 of all nonempty compact subsets of X × S 1 has the S-ECIP.

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.

We obtain a condition under which a uniform flow is induced over a metric space homeomorphic to a given metric space with a uniform flow. Various examples of uniform flow are also constructed.