FellowGiselle Monteiro
Project NameKurzweil's and Dobrakov's approaches to integration
Host organisationMathematical Institute
Duration of the project01.04.2016 - 31.03.2017

Abstract
When it comes to generalizations of the Lebesgue integral we can distinguish two different approaches: Riemann-type integrals and strong integrals. In this project we will deal with both approaches, being the focus of our attention the Kurzweil-Stieltjes and the Dobrakov integrals in abstract settings. Concerning the Kurzweil-Stieltjes integral - a nonabsolute integral whose definition is based on Riemann-type sums - we aim to investigate some particular properties (such as, boundedness and convergence) as well as the extension of its applicability to integration over more general sets. Moreover, a comparative analysis of Dobrakov and Kurzweil theories of integration is also part of our main goals.

Project Summary with Interim Results

The main scientific goal of this project was to describe the relationship between two type of integrals: one due to I. Dobrakov (a former fellow of the Mathematical Institute SAS), and the other developed by J. Kurzweil (from the Czech Academy of Sciences). Both notions generalize the known concept of Lebesgue integration, but while the former is an example of a strong integral, the latter is a non-absolute integral based on a simple modification of the Riemann integral. In order to investigate the connection between Dobrakov’s and Kurzweil’s approaches to integration, the intermediate objectives comprised two phases. The first corresponded to a continuation of the research on Kurzweil-Stieltjes integral as the analytical tool for the subsequent phase of the project. The second phase included the study of Dobrakov integral and a comparative analysis of Dobrakov and Kurzweil theories of integration.

The methodology employed in this project was the standard one used in mathematical academic research: bibliographical research, seminars, and the constructive mathematical thinking. Besides a consistent collaboration with the Mathematical Institute SAS, branch in Košice, consultations with Czech Academy of Sciences also played a significant role during this reporting period. A total of 18 lectures have been delivered at seminars and conferences, in Slovakia and abroad, to disseminate and promote the results obtained within this project.

In our research on Kurzweil-Stieltjes integral, we investigated whether this integral mimics some of the properties which are known for the Riemann-Stieltjes integral regarding some classes of functions. The integrability was then utilized to characterize functions of bounded variation and regulated functions. The results are contained in a paper accepted for publication. Other research questions on Kurzweil integration addressed during this period aimed at the improvement of the monograph whose pre-contract dates from 2014. The completion and publication of this collaborative work is expected in forthcoming months of 2017.

To undertake the proposed comparative analysis of Dobrakov’s and Kurzweil’s theories, a new notion of integral has been introduced, the m-integral. This new integral is defined with respect to a measure but its limiting process is based on gauge functions. Therefore, it incorporates important characteristics of both Dobrakov’s and Kurzweil’s approaches to integration. While the relation with abstract Kurzweil integral is straightforward, the identification between m-integral and Dobrakov integral is subject to some conditions. The study of the properties of the m-integral and its application to connect Dobrakov and Kurzweil integrals have been collected in a paper (to be submitted). 

Finally, besides pursuing the objectives stated, new collaborative research was set about in connection to the scientific visits undertook during the reporting period.  Dealing with applications of Kurzweil integration to equations, two papers are under preparation jointly with foreign institutions; one in Spain (University of Santiago de Compostela) and one in Poland (Adam Mickiewicz University).