Wavelet neural networks functional approximation and application

Approximation theory constitutes a useful field that is related to quasi all other fields, in both theoretical and applied sciences. In approximation theory, the aim is generally to construct an idea about a function that is usually impossible or difficult to evaluate directly, and which is usually unknown. Such functions appear widely in PDE, probability law distributions, statistical modeling, etc. Some of the most known approximators nowadays are neural networks and wavelets, which constitute good classes of elementary functions permitting as efficiently as possible to describe functions in appropriate spaces. This paper aims to develop combined neural networks and wavelet approximators for functions, based on the involvement of wavelets as activation functions. Some necessary conditions on the activation function to approximate [Formula: see text] and [Formula: see text]-elements are relaxed as well as those on the measure [Formula: see text]. We prove that for a wavelet activation function, any element of [Formula: see text] as well as [Formula: see text] can be well approximated for arbitrary measures [Formula: see text]. The theoretical results are subject to an experimental application in order to show their effectiveness.