My research interests focus on **dyadic harmonic analysis and its applications**.

Dyadic harmonic analysis is a branch of Fourier analysis which, instead of classic trigonometric series, studies series based on orthonormal systems composed by locally constant functions. Due to its apparent simplicity, the most well-known example of them is the *Walsh-Paley system*. This system is formed by Walsh functions which we know that they only take values 1 and -1. This property offers a wide range of applications in the world of digital technology. Indeed, the Walsh functions have a great advantage with respect to the classic trigonometric functions in the sense that computers can be very effective to determine the precise value of any Walsh function at any point. Currently, from the perspective of the digital technology the Walsh system fills the role as the classical trigonometric system in the analogue technology.

There are another known arrangements of Walsh functions, the *original Walsh systems* and the *Walsh-Kaczmarz system*. Here you can read a brief introduction about these systems and the relationship between them.

I implemented a package in Maple 15 computer algebra system with which researchers in dyadic harmonic analysis can easily and quickly perform calculations related to frequently used functions and operators in Fourier analysis with respect to the Walsh-Paley system. This packages can be used, inter alia, to confirm theoretical results, to search for counter-examples, to illustrate publications and presentations and for searching and confirming mathematical conjectures and requests. This package calculates very quickly operators for a large range of indices, since it is able to manage the discrete values of them as a vector. Here you can see the result of some Maple worksheet with the corresponding instructions to use the package properly ([1], [2] and [3]).

The Walsh functions can also be used for* solving differential equations numerically*. The method consists in discretizing the integral equation which is equivalent to the original differential equation, substituting the functions in it by its 2^{n}-th partial sums of Walsh-Fourier series and approaching the solution by a Walsh polynomial. In that way, the problem is reduced to solve a linear system. The method above was implemented in 1975 by C. F. Chen and C. H. Hsiao for the solution of ordinary linear systems of differential equations with constant coefficients. As a result of the joint work with György Gát, we made a deep analysis of the solution by proposing a new faster multistep method to obtain the same numerical solution (see our first publication in this topic) and also we extend the method for solving another types of differential equations.

In the recent decades, an increasing number of mathematicians supports the view that the appropriate environment for the development of Fourier analysis is the theory of locally compact groups. Fine proved the fact that Walsh functions can be represented on the dyadic group, i.e. the complete direct product of cyclic groups of order 2, as a system of characters. This meant that the dyadic analysis is included in the theory of the abstract harmonic analysis. This fact provided an opportunity to apply concepts, methods and results which were already known, and made it possible to draw a parallel between the digital models based on the continuous and Walsh functions.

The theory of dyadic analysis was enriched by the work of Vilenkin, when he generalized the Walsh-Paley system in 1947. He studied the commutative cases, the so-called Vilenkin systems which are formed by the character system of complete direct product of arbitrary cyclic groups.

Similarly, Vilenkin systems can be generalized considering the complete products of finite groups, where these finite groups are not necessarily commutative, and we follow the way of harmonic analysis to obtain the appropriate orthonormal systems. The properties of representative product systems can be very different relative to the properties of the Walsh-Paley and Vilenkin systems. For example they consist of not necessarily uniformly bounded functions, which may even take 0 as a value at some points. Therefore, the results we get will be different from the ones in the commutative theory. You can learn more about representative product systems in my PhD dissertation.