Generalized wavelet transforms over finite fields

ABSTRACT

In this article we introduce the abstract notion of generalized wavelet (affine) groups over finite fields as the finite group of generalized dilations, and translations. We then present a unified theoretical linear algebra approach to the theory of generalized wavelet transforms over finite fields. It is shown that each vector defined over a finite field can be represented as a finite coherent sum of generalized wavelet coefficients as well.

KEYWORDS:

• Finite field
• Galois group
• generalized wavelet (affine) group
• generalized wavelet representation
• generalized wavelet transform
• generalized dilation operator
• periodic (finite size) data
• prime integer

2010 MATHEMATICS SUBJECT CLASSIFICATIONS:

• Primary 42C40
• 12E20
• Secondary 13B05
• 12F10
• 81R05
• 20G40

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 $|G|$ denotes the order of the group G, or, more generally, the cardinality of a set G.