There are two main contributions in this paper. First, the \(X_{iw}\)-sum of Banach spaces is defined, following the work of Argyros and Motakis in [4]. In reality, this Banach space is an example of a \(U\)-sum of Banach spaces (e.g. in [12]), however, the alternate construction that is provided in this paper allows for a self-contained proof that it has the Lebesgue property if each summand has the Lebesgue property. This second result not only obtains a large new class of Banach spaces with the Lebesgue property, but it also shows that the non-asymptotic-\(\ell_{1}\) sum of infinite-dimensional Banach spaces still can have the Lebesgue property, which had not been established previously.

We consider the Volterra integration operator $V$, $Vf( x) =\int_{0}^{x} f( t)\mkern1mu dt$, on the following subspace of the Wiener algebra $W[ 0,1]$: \[ W^{( 1) }[ 0,1] :=\{ f\in W[ 0,1] : f^{\prime}\!\in W[ 0,1] \} . \] We investigate solvability of the operator equations $VA=\lambda AV$ and $VA=\lambda A^{2}V$, where $\lambda\in\mathbb{C}$ is a complex number. Our proof is based on the Duhamel product of functions defined by \[ ( f \circledast g) ( x) :=\frac{d}{dx}\int_{0}^{x}f( x-t) g( t) \, dt. \]

The purpose of this work is to examine the structure of optimal dual fusion frames and get more flexibility in the use of dual fusion frames for erasures of subspaces. We deal with optimal dual fusion frames with respect to different definitions of duality and compare the advantages of these approaches. In addition, we introduce a new concept so called partial optimal dual which involves less time and computation for detecting optimal dual for erasures in known locations. Then we study the relationship between local and global optimal duals by partial optimal duals which leads to some applicable results. In the sense that, we obtain an overcomplete frame and a family of associated optimal duals by a given Riesz fusion basis. We present some examples to exhibit the effect of error rate when dual fusion frames are applied in reconstruction.

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.

We study some variants of the Erdős similarity problem. We pose the question if every measurable subset of the real line with positive measure contains a similar copy of an infinite geometric progression. We construct a compact subset of the real line such that 0 is a Lebesgue density point of , but does not contain any (non-constant) infinite geometric progression. We give a sufficient density type condition that guarantees that a set contains an infinite geometric progression. By slightly improving a recent result of Bradford, Kohut and Mooroogen [1], we construct a closed set ⊂ [0, ∞) such that the measure of ∩ [ , + 1] tends to 1 at infinity but does not contain any infinite arithmetic progression. We also slightly improve a more general recent result by Kolountzakis and Papageorgiou [2] for more general sequences. We give a sufficient condition that guarantees that a given Cantor type set contains at least one infinite geometric progression with any quotient between 0 and 1. This can be applied to most symmetric Cantor sets of positive measure.

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 $1$.

I prove a mass transference principle for general shapes, similar to a recent result by H. Koivusalo and M. Rams. The proof relies on Vitali's covering lemma and manipulations with Riesz energies. The main novelty is that it is proved that the obtained limsup-set belongs to the classes of sets with large intersections, as defined by K. Falconer. This has previously not been proved for as general shapes as in this paper.