We consider aane systems in R n constructed from a given integral invertible and expansive matrix R, and a nite set B of translates, b x := R ?1 x + b; the corresponding measure on R n is a probability measure and xed by the selfsimilarity = jBj ?1 P b2B ?1 b. There are two a priori candidates for an associated orthogonal harmonic analysis : (i) the existence of some subset in R n such that the...

We discuss two affine inequalities and their role in two specific problems in harmonic analysis. The aim is to clarify the role that affine inequalities may play in harmonic analysis rather than to present any definitive results.

It is known that superpositions of ridge functions (single hidden-layer feedforward neural networks) may give good approximations to certain kinds of multivariate functions. It remains unclear, however, how to effectively obtain such approximations. In this paper, we use ideas from harmonic analysis to attack this question. We introduce a special admissibility condition for neural activation fu...

This thesis describes a system of neural networks that performs harmonic analysis on musical compositions. The system is split into four groups of Networks: Key Network, Root Network A (used to detect secondary functions), Root Network B, and Quality Network. The system is trained on 20 J.S. Bach chorales and tested on another 18 Bach chorales. Accuracies of over 90% were obtained for all four ...

In these lectures, we concentrate on the motivations, development and applications of the Calderon-Zygmund operator theory. Lecture 1. The differential operators with constant coefficients and the first generation of Calderon-Zygmund operators Consider the following differential operator with constant coefficients: Lu(x) = α a α ∂ α u ∂x α. By taking the Fourier transform, (Lu)(ξ) = α a α (−2πi...

The harmonic index H(G) , of a graph G is defined as the sum of weights 2/(deg(u)+deg(v)) of all edges in E(G), where deg (u) denotes the degree of a vertex u in V(G). In this paper we define the harmonic polynomial of G. We present explicit formula for the values of harmonic polynomial for several families of specific graphs and we find the lower and upper bound for harmonic index in Caterpill...