Abstract

The wavelet analysis of a function passes through its so-called wavelet transform. Such a transform is mathematically defined as a convolution product of the analyzed function with another analyzing function known as the mother wavelet by involving the scale and the translation parameters. This means that the mother wavelet construction is the starting and major point in the wavelet analysis. Besides, the choice of the mother wavelet remains a major problem in wavelet applications such as statistical series, time series, signal, and image processing. This needs more candidates of mother wavelets to be constructed. The main aim of the present paper is to construct a new mother wavelet by exploiting the well-known Farey map. We showed indeed that such a map may be a mother wavelet owing properties such as admissibility, moments, 2-scale relation, and reconstruction rule already necessary in the wavelet analysis of functions. By a suitable choice of translation-dilation parameters on the original Farey map, we succeeded to prove the main properties of a Farey wavelet analysis. The constructed mother looks to be suitable for many complicated applications such as hyperbolic PDEs.