This is a eulogy of Jaroslav Kurzweil who died on March 17, 2022. The author of important contributions on ordinary differential equations and the theory of integral, he was professor and researcher at the Institute of Mathematics of the Academy of Science and at Charles University in Prague.

We investigate variants of a three circles type theorem in the context of $\mathcal{Q}-$valued functions. We prove some convexity inequalities related to the $L^{2}$-growth function in the $\mathcal{Q}-$valued settings. Optimality of these inequalities and comparsion to the case of real valued harmonic functions is also discussed.

In this paper we will prove some properties of the algebras $W(I)$, $\;c^*(I)$ and $c(I)$ which arise when considering the notion of $I$-convergence of real sequences. By $I$-convergence of a real sequence we mean the convergence according to the filter associated to an ideal $I$ on the set $\mathbb{N}$. $W(I)$ is the set of sequences having finite $I$-variation, $ c^*$ and $c$ stand for two kinds of $I$-convergence. In particular, we are studying the maximal ideals of $W(I)$ in the cases where $I$ is admissible or non-admissible. We do that by using the topological properties of the semi-normed space $l^{\infty}(I)$. Finally, we discuss the question whether all maximal ideals $P$ of $W(I)$ are of the form $P=c_0(\mathcal{M})\cap W(I)$ where $\mathcal{M}$ is a maximal ideal of $2^{\mathbb{N}}$.

Using the Laplace derivative a Perron type integral, the Laplace integral, is defined. Moreover, it is shown that this integral includes Perron integral and to show that the inclusion is proper, an example of a function is constructed, which is Laplace integrable but not Perron integrable. Properties of integrals such as fundamental theorem of calculus, Hake's theorem, integration by parts, convergence theorems, mean value theorems, the integral remainder form of Taylor's theorem with an estimation of the remainder, are established. It turns out that concerning the Alexiewicz norm, the space of all Laplace integrable functions is incomplete and contains the set of all polynomials densely. Applications are shown to Poisson integral, a system of generalised ordinary differential equations and higher-order generalised ordinary differential equation.

Let $\widehat{\mu}$ be the Fourier transform of a Borel probability measure $\mu$ on $\mathbb R^d$. We look at the closed abelian subgroup $\Gamma (\mu)$ of $\mathbb R^d$, which consists of the periods of the function $\widehat{\mu}$. We prove the following dichotomy: i) The support of $\mu$ is non-degenerate if and only if $\Gamma (\mu)$ is a lattice. ii) The support of $\mu$ is degenerate if and only if $\Gamma (\mu)$ contains a linear subspace $\neq \{0\}$ of $\mathbb R^d$. A similar dichotomy is also discussed for the period group of the function $|\widehat{\mu}|$.

The basic idea we use throughout this paper is to generalize the construction of the Cantor set as the attractor of an iterated function system $\big(f_0,\ f_1\big)$, replacing the classical functions acting on $ [0,1]$ via $f_0(x) =\frac{x}{3}$ and $f_1(x)= \frac{2}{3} + \frac{x}{3}$ with the new functions acting via $f_0(x)= \theta x$ and $f_1(x) = 1 - \theta + \theta x$, where $\theta \in \Big[0,\ \frac{1}{2} \Big]$ is an arbitrary number. We add to the schema two positive numbers $p_0, \ p_1$ such that $p_0 + p_1 = 1$, obtaining the iterated function system with probabilities $\big(f_0,\; f_1 ; \; p_0,\; p_1\big)$ which generates the invariant set (attractor) and the invariant measure, according to J. Hutchinson. The structure of the invariant set (genera-lized Cantor set) being known, we focus on the detailed computation of the invariant measure. At the end of the paper, we study the dependence upon the parameter $\theta$ of the invariant set and of the invariant measure.

Using a novel approach, we develop a result analogous to the Lebesgue density theorem for Borel sets using the notion of category: If $B\subset \lbrack 0,1]$ is a Borel set, then there exists a first category set $S\subset B$ with the property that for every $x\in B-S$ there exists $\varepsilon >0$ such that $B\cap (x-\varepsilon ,x+\varepsilon )$ is a residual subset of $(x-\varepsilon ,x+\varepsilon )$.

We fill up some gaps in the existing literature on the Henstock-Kurzweil integration on metric measure spaces. The most important one is the choice of suitable candidates for `intervals' in metric spaces for which the conclusion of Cousin's lemma holds. We also provide some characterizations of compactness and completeness in terms of Cousin's lemma, along with some alternative proofs of a few related results. Then we improve the result regarding differentiability of the primitives of Henstock-Kurzweil integrable functions on metric spaces, and a few consequences. We propose shorter proofs of measure theoretic characterizations of the HK-integral in terms of the variational measure $V_F$ as well as the $ACG^\Delta$ functions. Finally, we present alternative proofs of some generalizations of two results on Lebesgue integration. The first one bypasses Vitali covering lemma for a result on absolute continuity, while the second one is a version of the fundamental theorem of calculus in our setting.

We introduce a non-autonomous generalization of spatial-temporal differentiations, and prove results about probabilistically and topologically generic behaviors of certain spatial-temporal differentations generated by endomorphisms of compact abelian metrizable groups.

We consider the class of all nonnegative on $\mathbb R_+$ functions such that each of them satisfies the reverse Hölder inequality uniformly over all intervals with some constant, the minimum value of which can be regarded as the corresponding “norm” of a function. We compare this “norm” with the “norm” of the even extension of a function from $\mathbb R_+$ on $\mathbb R$. In this paper, an upper estimate for the ratio of such “norms” has been obtained. For the special case of power functions on $\mathbb R_+$, we give the precise value of the “norm” increase caused by even extension. This value is a lower estimate for the analogous quantity in the case of arbitrary functions. It has been shown that the obtained upper and lower estimates for the general case are asymptotically sharp.

We prove that various geometrically significant sets are in fact null sets.

It is well-known that the generalized Riemann approach using non-uniform mesh has given rise to integrals which are more general than the Riemann integral. In this note, motivated by earlier studies on stochastic integrals, we consider special interval-point pair in defining the Riemann sum, where the point (or the tag) is the left-hand point of the interval. We show that this approach in fact is equivalent to the Lebesgue integral. Hence this simplifies McShane's construction of interval-point pair. Moreover, by restricting to the tag as the left hand point of the interval, the integration-by-substitution and by-parts formulae become easy consequences of the definition.

Sánchez, Viader, Paradís and Carrillo (2016) proved that there exists an increasing continuous singular function $f$ on $[0,1]$ such that the set $A_f$ of points where $f$ has a nonzero finite derivative has Hausdorff dimension 1 in each subinterval of $[0,1]$. We prove a stronger (and optimal) result showing that a set $A_f$ as above can contain any prescribed $F_{\sigma}$ null subset of $[0,1]$.

The statement of Theorem 1 in [1] contained an inadvertent error in its part (c). The correct statement of the theorem should read as follows.