Math page of Bartosz

Hi! My name is Bartosz, I am an assistant professor of mathematics at Mälardalen University in Sweden. Previously I was a lecturer in mathematics at KTH Royal Institute of Technology and a doctoral student at Lund University under the supervision of Alexandru Aleman.

I like mathematical analysis in general. My research interests include complex function theory, harmonic analysis and operator theory. In particular, I study approximation, cyclicity, boundary regularity, and Fourier uniqueness in spaces of analytic and harmonic functions. You can read more about my program and goals by clicking on Research in the menu.

Three Polish mathematicians hanging out: Otto Nikodym, me and Stefan Banach. Picture taken by Alberto Dayan!

Most of my research is concentrated in the areas of operator theory and harmonic analysis. The following three interconnected topics are particularly prominent in my work.

Hilbert spaces of analytic functions

The study of Hilbert spaces of analytic functions, such as Hardy spaces, Bergman spaces, Dirichlet spaces and de Branges-Rovnyak spaces H(b), is closely related to the fundamental shift operators. These operators translate a given sequence of numbers one step to the left or to the right. They are used in science and engineering to model the passage of time.

In the context of Hilbert spaces of analytic functions, the shift operators act by translating the Taylor series of an analytic function. We try to understand these operators better by studying, for instance, their cyclic vectors, their invariant subspaces, and the operator algebras that the shifts generate.

Example publications

Hilbert spaces of analytic functions with a contractive backward shift
With Alexandru Aleman, Journal of Functional Analysis (2019) doi
Universal Multipliers for Sub-Hardy Hilbert spaces
With Daniel Seco. To appear in Indiana University Mathematics Journal (2024) arXiv

Polynomial approximation in the plane

The famous Weierstrass Approximation Theorem says that the (algebraic) polynomials are uniformly dense among functions continuous on a finite interval. The result does not hold if we replace the interval by a circle, but in this context we have a celebrated result of Gábor Szegő. His result characterizes the weighted Lebesgue spaces on the circle in which the polynomials form a dense subset. It has important applications in prediction theory for stationary processes and in other areas of mathematics.

Less is known about polynomial approximation on other planar domains and other weighted Lebesgue spaces. My work in this direction is related to extensions of the theorem of Szegő to more general weighted Lebesgue spaces in the plane.

Example publications

Revisiting mean-square approximation by polynomials in the unit disk
Journal d'Analyse Mathématique (2025) doi
Thomson decompositions of measures in the disk
Transactions of the American Mathematical Society (2023) doi

Uniqueness problems in Fourier analysis

Surely one of the most intriguing predictions of quantum mechanics is the Heisenberg Uncertainty Principle which asserts a universal limitation to the accuracy of certain pairs of simultaneous measurements. The principle has a famous purely Fourier analytic formulation which is expressed in terms of a limitation of simultaneous concentration of the mass of a function and its frequency content (i.e., its Fourier transform).

In fact, there is a multitude of Fourier analytic theorems that prohibit simultaneous smallness or concentration of a function and its frequency content, in one way or another. Some of these statements find interesting interpretations in engineering. For instance, it is a consequence of a result in Fourier analysis that ideal band-pass filters do not exist.

A statement in Fourier analysis which prohibits the existence of an object satisfying simultaneously two properties, one in time domain and one in frequency domain, is often referred to as a uniqueness theorem. My work on uniqueness theorems is heavily inspired by applications in the theory of Hilbert spaces of analytic functions, polynomial approximations, and other parts of mathematics.

Example publications