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In this article, we establish theory of semi-orthogonal Parseval wavelets associated with generalized multiresolution analysis (GMRA) for local fields of positive characteristics (LFPC) and obtain their characterization in terms of consistency equation. As a consequence, we obtain a characterization of an orthonormal (multi)wavelet associated with an MRA in terms of multiplicity function as well as dimension function. Further, we provide characterizations of Parseval scaling functions, scaling sets and bandlimited wavelets together with a Shannon-type multiwavelet. Some examples of such wavelets are also produced for LFPC.

In this paper it is investigated as to when a nonempty invariant closed subset A of a -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 S-equivariant complete invariance property (S-ECIP). It is also shown that if X is a metric space and acts on by the action , where p, and , then the hyperspace of all nonempty compact subsets of has the S-ECIP.

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., Comment: arXiv admin note: text overlap with arXiv:1511.05703; Accepted in Mathematische Nachrichten, 2017