Local Fields Of Positive Characteristic Research Articles

Existence of unconditional wavelet bases for Lp-norm over a local fields of positive characteristic

In this paper, we defined Calder[Formula: see text]n–Zygmund operator [Formula: see text] as [Formula: see text] on local fields ([Formula: see text]) under certain conditions on [Formula: see text] and then find the boundedness of operator [Formula: see text]. Using result on boundedness of [Formula: see text] and orthonormal wavelet basis of [Formula: see text], we provide unconditional wavelet bases for [Formula: see text]-norm over a local fields of positive characteristic by the wavelet coefficients.

The Kakeya conjecture on local fields of positive characteristic

AbstractWe state and prove an analogue of the Kakeya conjecture for the local field . This extends Arsovski's result on the Kakeya conjecture to local fields of positive characteristic. We also prove the Kakeya maximal conjecture in this setting.

ON THE CHARACTERIZATION OF SCALING FUNCTIONS ON NON-ARCHEMEDEAN FIELDS

In real life application all signals are not obtained from uniform shifts; so there is a natural question regarding analysis and decompositions of these types of signals by a stable mathematical tool. This gap was filled by Gabardo and Nashed [11] by establishing a constructive algorithm based on the theory of spectral pairs for constructing non-uniform wavelet basis in \(L^2(\mathbb R)\). In this setting, the associated translation set \(\Lambda =\left\{ 0,r/N\right\}+2\,\mathbb Z\) is no longer a discrete subgroup of \(\mathbb R\) but a spectrum associated with a certain one-dimensional spectral pair and the associated dilation is an even positive integer related to the given spectral pair. In this paper, we characterize the scaling function for non-uniform multiresolution analysis on local fields of positive characteristic (LFPC). Some properties of wavelet scaling function associated with non-uniform multiresolution analysis (NUMRA) on LFPC are also established.

Frame multiresolution analysis in discrete settings on local fields of positive characteristic

Frame multiresolution analysis in discrete settings on local fields of positive characteristic

A Short Note on Wavelet Frames Based on FMRA on Local Fields

The concept of frame multiresolution analysis (FMRA) on local fields of positive characteristic was given by Shah in his paper, Frame Multiresolution Analysis on Local Fields published by Journal of Operators. The author has studied the concept of minimum-energy wavelet frames on these prime characteristic fields. We continued the studies based on frame multiresolution analysis and minimum-energy wavelet frames on local fields of positive characteristic. In this paper, we introduce the notion of the construction of minimum-energy wavelet frames based on FMRA on local fields of positive characteristic. We provide a constructive algorithm for the existence of the minimum-energy wavelet frame on the local field of positive characteristic. An explicit construction of the frames and bases is given. In the end, we exhibit an example to illustrate our algorithm.

Discrete orthogonal and Riesz refinable functions on local fields of positive characteristic

Necessary conditions under which shift systems of refinable functions are orthogonal, or Riesz systems on Vilenkin groups and local fields of positive characteristic, are well known. These conditions are expressed in terms of a mask. We indicate a class of refinable functions for which these conditions are necessary and sufficient.

Multiplicity one theorem for $$(\mathrm {GL}_{n+1},\mathrm {GL}_n)$$ over a local field of positive characteristic

Let $\mathbb{F}$ be a non-archimedean local field of positive characteristic different from 2. We consider distributions on $\mathrm{GL}(n+1,\mathbb{F})$ which are invariant under the adjoint action of $\mathrm{GL}(n,\mathbb{F})$. We prove that any such distribution is invariant with respect to transposition. This implies that the restriction to $\mathrm{GL}(n,\mathbb{F})$ of any irreducible smooth representation of $\mathrm{GL}(n+1,\mathbb{F})$ is multiplicity free.

Dualwavelet frames in Sobolev spaces on local fields of positive characteristic

Wavelet frames have gained considerable popularity during the past decade, primarily due to their substantiated applications in diverse and widespread fields of engineering and science. In this article, we obtain the characterization of nonhomogeneous wavelet frames and nonhomogeneous dual wavelet frames in a Sobolev spaces on a local field of positive characteristic by means of a pair of equations.

Contragredient representations over local fields of positive characteristic

It is conjectured by Adams-Vogan and Prasad that under the local Langlands correspondence, the L-parameter of the contragredient representation equals that of the original representation composed with the Chevalley involution of the L-group. We verify a variant of their prediction for all connected reductive groups over local fields of positive characteristic, in terms of the local Langlands parameterization of Genestier-Lafforgue. We deduce this from a global result for cuspidal automorphic representations over function fields, which is in turn based on a description of the transposes of V. Lafforgue's excursion operators.

Super‐wavelets on local fields of positive characteristic

AbstractThe 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.

Equidistribution with an error rate and Diophantine approximation over a local field of positive characteristic

For a local field K of formal Laurent series and its ring Z of polynomials, we prove a pointwise equidistribution with an error rate of each H-orbit in SL(d, K)/SL(d, Z) for a certain proper subgroup H of a horospherical group, extending a work of Kleinbock-Shi-Weiss. We obtain an asymptotic formula for the number of integral solutions to the Diophantine inequalities with weights, generalizing a result of Dodson-Kristensen-Levesley. This result enables us to show pointwise equidistribution for unbounded functions of class Cα.

Polyphase matrix characterization of framelets on local fields of positive characteristic

Abstract An important tool for the construction of framelets on local fields of positive characteristic using unitary extension principle was presented by Shah and Debnath [Tight wavelet frames on local fields, Analysis, 33 (2013), 293-307]. In this article, we continue the study of framelets on local fields and present a polyphase matrix characterization of framelets generated by the extension principle.

How to construct wavelets on local fields of positive characteristic

We present an algorithm for construction step wavelets on local fields of positive characteristic.

Affine, quasi-affine and co-affine frames on local fields of positive characteristic

The concept of quasi-affine frame in Euclidean spaces was introduced to obtain translation invariance of the discrete wavelet transform. We extend this concept to a local field K of positive characteristic. We show that the affine system generated by a finite number of functions is an affine frame if and only if the corresponding quasi-affine system is a quasi-affine frame. In such a case the exact frame bounds are equal. This result is obtained by using the properties of an operator associated with two such affine systems. We characterize the translation invariance of such an operator. A related concept is that of co-affine system. We show that there do not exist any co-affine frame in L2(K).

Fast discrete Fourier transform on local fields of positive characteristic

For the discrete Fourier transform with respect to the system of characters of a local field with positive characteristic, we propose a fast algorithm. We find the complexity of the algorithm.

Nonuniform wavelet packets on local fields of positive characteristic

Abstract The concept of nonuniform multiresolution analysis on local field of positive characteristic was considered by Shah and Abdullah for which the translation set is a discrete set which is not a group. We construct the associated wavelet packets for such an MRA and investigate their properties by means of the Fourier transform.

On orthogonal systems of shifts of scaling function on local fields of positive characteristic

We present a new method for constructing an orthogonal step scaling function on local fields of positive characteristic, which generates multiresolution analysis.

Gabor frames on local fields of positive characteristic

Gabor frames have gained considerable popularity during the past decade, primarily due to their substantiated applications in diverse and widespread fields of engineering and science. Finding general and verifiable conditions which imply that the Gabor systems are Gabor frames is among the core problems in time-frequency analysis. In this paper, we give some simple and sufficient conditions that ensure a Gabor system $\left\{M_{u(m)b}T_{u(n)a}g=:\chi_{m}(bx)g\big(x-u(n)a\big)\right\}_{ m,n\in\mathbb N_0}$ to be a frame for $L^2(K)$. The conditions proposed are stated in terms of the Fourier transforms of the Gabor system's generating functions.

Construction of biorthogonal wavelet packets on local fields of positive characteristic

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.

Non-Haar MRA on local fields of positive characteristic

We propose a simple method to construct integral periodic mask and corresponding scaling step functions that generate non-Haar orthogonal MRA on the local field F(s) of positive characteristic p. To construct this mask we use two new ideas. First, we consider local field as vector space over the finite field GF(ps). Second, we construct scaling function by arbitrary tree that has ps vertices. By fixed prime number p there exist ps(ps−2) such trees.