In this paper, we present a unified approach to the study of shift-invariant systems to be frames on local fields of positive characteristic. We establish a necessary condition and three sufficient conditions under which the shift-invariant systems on local fields constitute frames for L2(K). As an application of these results, we obtain some known conclusions about the Gabor frames and wavelet frames on local fields.

Frames associated with shift-invariant spaces on local fields

Two sufficient conditions in frequency domain for Gabor frames

Affine frame is a very basic concept in wavelet theory. In the well-known Daubechies Criterion for affine frames the quantities which are calculated in terms of absolute values | (aJω) (aJω + lT)| are applied. In the new criterion established in this paper the authors use new criterion quantities which are calculated via algebric sums of suitable combinations of (ajω) (ajω + lT). When is an even function, the new criterion is more advanced than Daubechies criterion.

A new criterion on affine frames

Some characterizations of the multivariate wave packet systems in terms of the Fourier transforms of the wave packet systems' generating functions are discussed in this paper. First, the characterizations of the orthogonal wave packet systems are provided. Secondly, a sufficient condition of the completeness of wave packet system in some special cases is established. Finally, the necessary conditions and sufficient conditions for the wave packet systems to be wave packet Parseval frames with the very general lattices are obtained. Thus, the corresponding known results in Gabor systems and wavelet systems are obtained as some corollaries.

/pdf/characterizations-of-the-multivariate-wave-packet-systems-arfllvcwx5.pdf

Characterizations of the multivariate wave packet systems

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.

Gabor Frames on Local Fields of Positive Characteristic

Abstract The main purpose of this paper is to establish a necessary condition and two sufficient conditions ensuring that the shift-invariant system is a frame for . As some applications, the results are used to obtain some known conclusions about wavelet frames and Gabor frames.

Unified conditions for wavelet frames

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Unified conditions for wavelet frames