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

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.; Comment: 21 pages

Multiplicatively invariant (MI) spaces are closed subspaces of $L^2(\Omega,\mathcal{H})$ that are invariant under multiplications of (some) functions in $L^{\infty}(\Omega)$. In this paper we work with MI spaces that are finitely generated. We prove that almost every linear combination of the generators of a finitely generated MI space produces a new set on generators for the same space and we give necessary and sufficient conditions on the linear combinations to preserve frame properties. We then apply what we prove for MI spaces to system of translates in the context of locally compact abelian groups and we obtain results that extend those previously proven for systems of integer translates in $L^2(\mathbb{R}^d)$.; Comment: 13 pages. Minor changes have been made. To appear in Studia Mathematica

Orthogonal wavelet packets lack symmetry which is a much desired property in image and signal processing. The biorthogonal wavelet packets achieve symmetry where the orthogonality is replaced by the biorthogonality. In the present paper, we construct biorthogonal wavelet packets on local fields of positive characteristic and investigate their properties by means of the Fourier transforms. We also show how to obtain several new Riesz bases of the space L2(K) by constructing a series of subspaces of these wavelet packets. Finally, we provide the algorithms for the decomposition and reconstruction using these biorthogonal wavelet packets.

We provide a characterization of wavelets on local fields of positive characteristic based on results on affine and quasi affine frames. This result generalizes the characterization of wavelets on Euclidean spaces by means of two basic equations. We also give another characterization of wavelets. Further, all wavelets which are associated with a multiresolution analysis on a such a local field are also characterized.; Comment: arXiv admin note: text overlap with arXiv:1103.0090