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

Abstract

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).

Frames associated with shift-invariant spaces on local fields

“…There is some sort of equivalence between wavelet and quasi-wavelet systems. Indeed, Behera and Jehan [6] proved in full generality the following result on local fields of positive characteristic. For j ∈ N 0 , let N j denotes a full collection of coset representatives of N 0 /q j N 0 , i.e.,…”

Shift-invariant subspaces and wavelets on local fields

“…For some aspects of the wavelet theory on local fields of positive characteristic, we refer to [1][2][3][4][5][6]11,14,15].…”

Dilation Operators in Besov Spaces over Local Fields

“…Jiang, Li and Jin [5] have introduced the concept of multiresolution analysis and wavelet frames on local fields [6]. Later, Behera and Jahan have developed the theory of wavelets on such a field in a series of papers [1,2,3].…”