Remarks on dyadic analysis

About 100 years ago, N.J. Walsh's fundamental paper [25] was published, in which he introduced the digital version of the trigonometric system. In remembrance of this and the 50th anniversary of Walsh's death, the authors of the paper [2] presented the role of Walsh functions in dyadic analysis and technical applications. 35 years ago, a collaboration between researchers from the Department of Numerical Analysis, Eötvös Loránd University and Professor W.R. Wade (University of Tennessee, USA) resulted in the publication of the first monograph on dyadic analysis. This provided an overview of the significant results in the field before 1990. Since then, several promising results have been achieved that may determine the future direction of research. This paper provides a brief overview of these results. In the commemorations prepared for the anniversary of the department's establishment, we present in detail our contributions to the achievements in the field. Here, the author only highlights the following. Regarding the Vilenkin generalization of the Walsh system, the interpretation of the concept of the conjugate function and the proof of the corresponding fundamental inequalities were significant [21]. The author of the [11] paper introduced the dyadic analogues of Hermite functions as eigenfunctions of the dyadic derivative and pointed out their application possibilities. In harmonic analysis, the examination of multiplier operators and the corresponding filtering procedures in signal processing is a central theme of research. The strong approximation, two-sided Sidon-type inequalities, and Hardy-type spaces related to this have proven to be of fundamental importance in both the trigonometric and dyadic cases [4]. It would be worthwhile to extend these results to Malmquist–Takenaka systems. New, significant results have also been achieved in the extension of multivariable dyadic analysis, traditional stochastic structures, and function spaces [26, 27]. The [6] paper provides insights into the studies related to the direct product of finite, non-commutative groups.