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

TL;DR

This paper extends the concept of quasi-affine frames to local fields of positive characteristic, establishing conditions for affine frames and showing the non-existence of co-affine frames in this setting.

Contribution

It generalizes quasi-affine frames to local fields of positive characteristic and characterizes affine frames via operator properties, also proving no co-affine frames exist.

Findings

Affine systems form frames iff quasi-affine systems do, with equal bounds.

Operator properties characterize translation invariance.

No co-affine frames exist in $L^2(K)$.

Abstract

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 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 $L^2(K)$.