Local Fields Of Positive Characteristic Research Articles

Vector-valued nonuniform multiresolution analysis on local fields

A multiresolution analysis (MRA) on local fields of positive characteristic was defined by Shah and Abdullah for which the translation set is a discrete set which is not a group. In this paper, we continue the study based on this nonstandard setting and introduce vector-valued nonuniform multiresolution analysis (VNUMRA) where the associated subspace V0 of L2(K, ℂM) has an orthonormal basis of the form {Φ (x - λ)}λ∈Λ where Λ = {0, r/N} + 𝒵, N ≥ 1 is an integer and r is an odd integer such that r and N are relatively prime and 𝒵 = {u(n) : n ∈ ℕ0}. We establish a necessary and sufficient condition for the existence of associated wavelets and derive an algorithm for the construction of VNUMRA on local fields starting from a vector refinement mask G(ξ) with appropriate conditions. Further, these results also hold for Cantor and Vilenkin groups.

Tight framelet packets on local fields of positive characteristic

Tight framelet packets on local fields of positive characteristic

Wave packet frames on local fields of positive characteristic

The objective of this paper is to construct wave packet frames on local fields of positive characteristic. A necessary and sufficient condition for the wave packet system DpjTu(n)aEu(m)bψj∈Z,m,n∈N0 to be a frame for L2(K) is given by means of the Fourier transform.

A characterization of tight wavelet frames on local fields of positive characteristic

The objective of this paper is to establish a complete characterization of tight wavelet frames on local fields of positive characteristic by means of two basic equations in the Fourier domain.

Nonuniform Multiresolution Analysis on Local Fields of Positive Characteristic

A generalization of Mallat’s classic theory of multiresolution analysis (MRA) on local fields of positive characteristic was considered by Jiang et al. (J Math Anal Appl 294:523–532, 2004). In this paper, we present a notion of nonuniform MRA on local field \(K\) of positive characteristic. The associated subspace \(V_0\) of \(L^2(K)\) has an orthonormal basis, a collection of translates of the scaling function \(\varphi \) of the form \(\{ \varphi (x-\lambda ) \}_{ \lambda \in \Lambda }\) where \(\Lambda = \{ 0,r/N \}+ \mathcal{Z}, \,N \ge 1\) is an integer and \(r\) is an odd integer such that \(r\) and \(N\) are relatively prime and \(\mathcal{Z}=\{u(n): n\in \mathbb {N}_{0}\}\). We obtain the necessary and sufficient condition for the existence of associated wavelets and present an algorithm for the construction of nonuniform MRA on local fields starting from a low-pass filter \(m_{0}\) with appropriate conditions.

Characterization of wavelets and MRA wavelets on local fields of positive characteristic

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 such a local field are also characterized.

Haar wavelets on the Lebesgue spaces of local fields of positive characteristic

Haar wavelets on the Lebesgue spaces of local fields of positive characteristic

Wavelet packets and wavelet frame packets on local fields of positive characteristic

Using a prime element of a local field K of positive characteristic, the concepts of multiresolution analysis (MRA) and wavelets can be generalized to such a field. We prove a version of the splitting lemma for this setup and using this lemma we construct the wavelet packets associated with such MRAs. We show that these wavelet packets generate an orthonormal basis by translations only. We also prove an analogue of the splitting lemma for frames and construct the wavelet frame packets in this setting.

Locally analytic functions in a local field of positive characteristic

In a recent paper (Buium et al., 2011 [3] ), Buium et al. proved that f is a locally analytic function from the p -adic integers, Z p to itself if and only if it is written as a restricted power series over Z p with finitely many iterates of the Fermat quotient operator. In this paper we establish the function field analogs of their result. In addition, we deduce Wagnerʼs result, which is the function field analog of Mahlerʼs result for continuous functions on Z p .

Lattices of minimum covolume in Chevalley groups over local fields of positive characteristic

In this article, we show that if G is a simply connected Chevalley group of either classical type of rank bigger than 1 or type E6, and q > 9 is a power of a prime number p > 5, then G = G(F_q((1/t))), up to an automorphism, has a unique lattice of minimum covolume, which is G(F_q[t]).

Interpolation of numbers of Catalan type in a local field of positive characteristic

Let $k$ be a local field of positive characteristic. Let $L$ be a cocompact discrete subgroup of $k$. Let $U$ be an open compact subgroup of $k$. Let $\ell$, $u$ and $a$ be elements of $k$, with $a$ nonzero. We study the behavior of the product $\prod_{0\neq x\in (\ell+L)\cap a(u+U)}x$ as $a$ varies. Our main result, which blends harmonic analysis and local class field theory, provides local confirmation of a conjecture concerning global function fields recently made by the author (math.NT/0407535).

Metric Diophantine approximation over a local field of positive characteristic

The conjectures of Sprindžuk in the metric theory of Diophantine approximation are established over a local field of positive characteristic. In the real case, these were settled by D. Kleinbock and G.A. Margulis using a new technique which involved nondivergence estimates for quasi-polynomial flows on the space of lattices. We extend their technique to the positive characteristic setting.