Professor Jaroslav Smítal passed away at the age of 79 years on March 9th, 2022. He was the founder of Czech-Slovak school of discrete dynamical systems and we all will miss him not only as a great mathematician and an extraordinary member of the Czech mathematical community, but above all as a good friend, colleague and a very inspiring person. The Czech-Slovak Workshop on Discrete Dynamical Systems 2022 was devoted to his memory and to celebrate his life's work. This article consists of personal recollections composed by several different authors, Jaro's colleagues and friends.

Jaroslav Smítal was a distinguished Czech and Slovak mathematician. The first part of the article is devoted to important moments of his life. In the second part we briefly discuss some of his scientific results.

This survey paper is based on a talk given at the 44th Summer Symposium in Real Analysis in Paris.

This line of research was initiated by a question of Haight and Weizsäker concerning almost everywhere convergence properties of series of the form $\sum_{n=1}^{{\infty}}f(nx)$.

A more general, additive version of this problem is the following:

Suppose $\Lambda$ is a discrete infinite set of nonnegative real numbers. We say that $ {\Lambda}$ is of type 1 if the series $s(x)=\sum_{\lambda\in\Lambda}f(x+\lambda)$ satisfies a zero-one law. This means that for any non-negative measurable $f: \mathbb R\to [0,+ {\infty})$ either the convergence set $C(f, {\Lambda})=\{x: s(x)<+ {\infty} \}= \mathbb R$ modulo sets of Lebesgue zero, or its complement the divergence set $D(f, {\Lambda})=\{x: s(x)=+ {\infty} \}= \mathbb R$ modulo sets of measure zero. If $ {\Lambda}$ is not of type 1 we say that $ {\Lambda}$ is of type 2.

The exact characterization of type $1$ and type $2$ sets is still not known.

The part of the paper discussing results concerning this question is based on several joint papers written at the beginning with J-P. Kahane and D. Mauldin, later with B. Hanson, B. Maga and G. Vértesy.

Apart from results from the above project we also cover historic background, other related results and open questions.

We generalize the well-known equality that the limit of the $L^p$ norm of a function as $p\rightarrow\infty$ is the $L^\infty$ norm to the scale of Orlicz spaces.

We prove that every finite Borel measure \( \mu \) in \( \mathbb R^N \) that is bounded from above by the Hausdorff measure \( \mathcal{H}^s \) can be split in countable many parts \( \mu{\lfloor_{_{{E_k}}}} \) that are bounded from above by the Hausdorff content \( \mathcal{H}^s_\infty \). Such a result generalises a theorem due to R. Delaware that says that any Borel set with finite Hausdorff measure can be decomposed as a countable disjoint union of straight sets. We apply this decomposition to give a simpler proof for the existence of solutions of a Dirichlet problem involving an exponential nonlinearity.

In this paper, we answer a question raised by David H. Fremlin about the Hausdorff measure of $\mathbb R^2$ with respect to a distance inducing the Euclidean topology. In particular we prove that the Hausdorff $n$-dimensional measure of $\mathbb R^n$ is never $0$ when considering a distance inducing the Euclidean topology. Finally, we show via counterexamples that the previous result does not hold in general if we remove the assumption on the topology.

In the present paper we discuss the rate of the approximation by the matrix transform of special partial sums of Walsh-Fourier series in $L_p(G)$ space ($1\leq p \lt \infty$) and in $C(G)$.

In this note, we give criteria for the irreducibility of functions in $ C^{m}\left[ 0,1\right] $, where $ m \in \{1,2,3,...\}\cup\{\infty\} \cup \{\omega\} $. We also discuss irreducibility in multivariable differentiability classes. Moreover, we characterize irreducible functions and maximal ideals in $ C^{\infty}\left[ 0,1\right] $. In fact, irreducible and prime smooth functions are the same, and every maximal ideal of $ C^{\infty}\left[ 0,1\right] $ is principal.

We prove that nondegenerate singular integral operators of convolution type are bounded on a rearrangement-invariant Banach function space $X(\mathbb{R}^d)$ if and only if its Boyd indices are nontrivial, extending the result by David Boyd (1966) for the Hilbert transform.

We study “distance spheres”: the set of points lying at constant distance from a fixed arbitrary subset $K$ of $[0,1]^d$. We show that, away from the regions where $K$ is “too dense” and a set of small volume, we can decompose $[0,1]^d$ into a finite number of sets on which the distance spheres can be “straightened” into subsets of parallel $(d-1)$-dimensional planes by a bi-Lipschitz map. Importantly, the number of sets and the bi-Lipschitz constants are independent of the set $K$.

The differentiability of the one parameter family of Okomoto's functions as functions of $x$ has been analyzed extensively since their introduction in 2005. As an analogue to a similar investigation, in this paper, we consider the partial derivative of Okomoto's functions with respect to the parameter $a$. We place a significant focus on $a = 1/3$ to describe the properties of a nowhere differentiable function $K(x)$ for which the set of points of infinite derivative produces an example of a measure zero set with Hausdorff dimension $1$.

The Dominated Convergence Theorem, Fatou's Lemma, and the Monotone Convergence Theorem are collectively referred to as the continuity theorems of Lebesgue integration. These ubiquitous results are stated and proved in the literature for sequences of measurable functions. It is well known that these results do not hold for arbitrary nets of measurable functions. These results, as well as the Levy Continuity Theorem, Riesz's characterization of $\mathcal{L}_p$ convergence in terms of almost everywhere convergence and convergence of norms, and a stronger version of Scheffe's Theorem, are derived for nets indexed by countably accessible directed sets.

We discuss and verify an interesting method for evaluating certain integrals of the form $\int_0^\infty f(t)\,dt$ using Laplace transforms. Careful proofs are given to justify the method and establish some limits that streamline the computations. Several examples are provided to illustrate the ease with which this method computes the value of an integral as compared to other standard approaches.