arXiv subjects

Paweł Pasteczka

Publications and source records attributed to Paweł Pasteczka.

At least 37 records · Page 2Linked to original sources

arXiv2021

On the Hardy property of mixed means

Hardy property of means has been extensively studied by Páles and Pasteczka since 2016. The core of this research is based on few of their properties: concavity, symmetry, monotonicity, repetition invariance and homogeneity (last axiom was recently omitted using some homogenizations techniques). In the present paper we deliver a study of possible omitting monotonicity and replacing repetition invariance by a weaker axiom. These results are then used to establish the Hardy constant for certain types of mixed means.

math.CA↗

arXiv2021

On the integral approach to means and their Hardy property

The celebrated Hardy inequality can be written in the form $$\int_0^\infty \mathcal{P}_p \big(f|_{[0,x]}\big)dx \le (1-p)^{-1/p} \int_0^\infty f(x)\:dx \qquad \text{ for }p\in(0,1)\text{ and }f \in L^1\text{ with }f\ge0,$$ where $\mathcal{P}_p$ stands for the $p$-th power mean. One can ask about possible generalizations of this property to another families (with sharp constant depending on the mean). Adapting the notion of Riemann integral, for every weighted mean we define the lower and the upper integral mean. We prove that every symmetric, monotone, $\mathbb{R}$-weighted mean on $I$ which is continuous in its entries and weights has at most one continuous extension to the integral one. Moreover this extension preserves the Hardy constant. This result allows to extend the latter inequality to the family of homogeneous, concave deviation means.

math.CA↗

arXiv2021

Error recognition in the Cantor cube

Based on the notion of thin sets introduced recently by T.~Banakh, Sz.~Głąb, E.~Jabłońska and J.~Swaczyna we deliver a study of the infinite single-message transmission protocols. Such protocols are associated with a set of admissible messages (i.e. subsets of the Cantor cube $\mathbb{Z}_2^ω$). Using Banach-Mazur games we prove that all protocols detecting errors are Baire spaces and generic (in particular maximal) ones are not neither Borel nor meager. We also show that the Cantor cube can be decomposed to two thin sets which can be considered as the infinite counterpart of the parity bit. This result is related to so-called xor-sets defined by D.~Niwiński and E.~Kopczyński in 2014.

math.LO↗

arXiv2020

On the $L^1$ and pointwise divergence of continuous functions

For a family of continuous functions $f_1,f_2,\dots \colon I \to \mathbb{R}$ ($I$ is a fixed interval) with $f_1\le f_2\le \dots$ define a set $$ I_f:=\big\{x \in I \colon \lim_{n \to \infty} f_n(x)=+\infty\big\}.$$ We study the properties of the family of all admissible $I_f$-s and the family of all admissible $I_f$-s under the additional assumption $$ \lim_{n \to \infty} \int_x^y f_n(t)\:dt=+\infty \quad \text{ for all }x,y \in I\text{ with }x<y.$$ The origin of this problem is the limit behaviour of quasiarithmetic means.

math.CA↗

arXiv2020

On the Jensen convex and Jensen concave envelopes of means

In recent papers the convexity of quasiarithmetic means was characterized under twice differentiability assumptions. One of the main goals of this paper is to show that the convexity or concavity of a quasiarithmetic mean implies the the twice continuous differentiability of its generator. As a consequence of this result, we can characterize those quasiarithmetic means which admit a lower convex and upper concave quasiarithmetic envelop.

math.CA↗

arXiv2020

Decision making via generalized Bajraktarević means

We define decision-making functions which arise from studying the multidimensional generalization of the weighted Bajraktarević means. It allows a nonlinear approach to optimization problems. These functions admit several interesting (from the point of view of decision-making) properties, for example, delegativity (which states that each subgroup of decision-makers can aggregate their decisions and efforts), casuativity (each decision affects the final outcome except two trivial cases) and convexity-type properties. Beyond establishing the most important properties of such means, we solve their equality problem, we introduce a notion of synergy and characterize the null-synergy decision-making functions of this type.

math.OC↗

arXiv2020

Invariant means, complementary averages of means, and a characterization of the beta-type means

We prove that whenever the selfmapping $(M_1,\dots,M_p)\colon I^p \to I^p$, ($p \in \mathbb{N}$ and $M_i$-s are $p$-variable means on the interval $I$) is invariant with respect to some continuous and strictly monotone mean $K \colon I^p \to I$ then for every nonempty subset $S \subseteq\{1,\dots,p\}$ there exists a uniquely determined mean $K_S \colon I^p \to I$ such that the mean-type mapping $(N_1,\dots,N_p) \colon I^p \to I^p$ is $K$-invariant, where $N_i:=K_S$ for $i \in S$ and $N_i:=M_i$ otherwise. Moreover \begin{equation*} \min(M_i\colon i \in S)\le K_S\le \max(M_i\colon i \in S). \end{equation*} Later we use this result to: (1) construct a broad family of $K$-invariant mean-type mappings, (2) solve functional equations of invariant-type, and (3) characterize Beta-type means.

math.CA↗

arXiv2020

Mean-type mappings and invariance principle

In the finite dimensional case, mean-type mappings, their invariant means, relations between the uniqueness of invariant means and convergence of orbits of the mapping, are considered. In particular it is shown, that the uniqueness of an invariance mean implies the convergence of all orbits. A strongly irregular mean-type mapping is constructed and its unique invariant mean is determined. An application in solving a functional equation is presented.

math.CA↗

arXiv2020

Quasiarithmetic-type invariant means on probability space

For a family $(\mathscr{A}_x)_{x \in (0,1)}$ of integral quasiarithmetic means sattisfying certain measurability-type assumptions we search for an integral mean $K$ such that $K\big((\mathscr{A}_x(\mathbb{P}))_{x \in (0,1)}\big)=K(\mathbb{P})$ for every compactly supported probabilistic Borel measure $\mathbb{P}$. Also some results concerning the uniqueness of invariant means will be given.

math.FA↗

arXiv2019

Online premeans and their computation complexity

We extend some approach to a family of symmetric means (i.e. symmetric functions $\mathscr{M} \colon \bigcup_{n=1}^\infty I^n \to I$ with $\min\le \mathscr{M}\le \max$; $I$ is an interval). Namely, it is known that every symmetric mean can be written in a form $\mathscr{M}(x_1,\dots,x_n):=F(f(x_1)+\cdots+f(x_n))$, where $f \colon I \to G$ and $F \colon G \to I$ ($G$ is a commutative semigroup). For $G=\mathbb{R}^k$ or $G=\mathbb{R}^k \times \mathbb{Z}$ ($k \in \mathbb{N}$) and continuous functions $f$ and $F$ we obtain two series of families (depending on $k$). It can be treated as a measure of complexity in a family of means (this idea is inspired by theory of regular languages and algorithmics). As a result we characterize celebrated families of quasi-arithmetic means ($G=\mathbb{R}\times \mathbb{Z}$) and Bajraktarević means ($G=\mathbb{R}^2$ under some additional assumptions). Moreover, we establish certain estimations of complexity for several other classical families.

math.GM↗

arXiv2019

New bounds for the ratio of power means

We show that for real numbers $p,\,q$ with $q<p$, and the related power means $\mathscr{P}_p$, $\mathscr{P}_q$, the inequality $$\frac{\mathscr{P}_p(x)}{\mathscr{P}_q(x)} \le \exp \bigg( \frac{p-q}8 \cdot \bigg(\ln\bigg(\frac{\max x}{\min x}\bigg)\bigg)^2 \:\bigg)$$ holds for every vector $x$ of positive reals. Moreover we prove that, for all such pairs $(p,q)$, the constant $\tfrac{p-q}8$ is sharp.

math.CA↗

arXiv2019

On Hardy type inequalities for weighted quasideviation means

Using recent results concerning the homogenization and the Hardy property of weighted means, we establish sharp Hardy constants for concave and monotone weighted quasideviation means and for a few particular subclasses of this broad family. More precisely, for a mean $\mathscr{D}$ like above and a sequence $(λ_n)$ of positive weights such that $λ_n/(λ_1+\dots+λ_n)$ is nondecreasing, we determine the smallest number $H \in (1,+\infty]$ such that $$ \sum_{n=1}^\infty λ_n \mathscr{D}\big((x_1,\dots,x_n),(λ_1,\dots,λ_n)\big) \le H \cdot \sum_{n=1}^\infty λ_n x_n \text{ for all }x \in \ell_1(λ). $$ It turns out that $H$ depends only on the limit of the sequence $(λ_n/(λ_1+\dots+λ_n))$ and the behaviour of the mean $\mathscr{D}$ near zero.

math.CA↗

arXiv2019

From the Ingham--Jessen property to mixed-mean inequalities

For every symmetric mean $\mathscr{M} \colon \bigcup_{n=1}^\infty I^n \to I$ (where $I$ an interval) and a nonzero function $W \colon \{1,\dots,n\} \to \mathbb{N} \cup \{0\}$, define an $n$-variable mean by $$\mathscr{M}_W(x):=\mathscr{M}\big(\underbrace{x_1,\dots,x_1}_{W(1)\text{-times}},\dots,\underbrace{x_n,\dots,x_n}_{W(n)\text{-times}}\big) \text{ for }x=(x_1,\dots,x_n) \in I^n.$$ Given two symmetric means $\mathscr{M},\,\mathscr{N} \colon \bigcup_{n=1}^\infty I^n \to I$ satisfying the so-called Ingham--Jessen inequality and some nonzero functions $F_1,\dots,F_k$, $G_1,\dots,G_l \colon \{1,\dots,n\} \to \mathbb{N} \cup \{0\}$, we establish sufficient conditions for inequalities of the form $$\mathscr{N} \big( \mathscr{M}_{F_1}(x),\dots,\mathscr{M}_{F_k}(x)\big) \le \mathscr{M} \big( \mathscr{N}_{G_1}(x),\dots,\mathscr{N}_{G_l}(x)\big) \qquad(x \in I^n).$$ Our results provide a unified approach to the celebrated inequalities obtained by Kedlaya in 1994 and by Leng--Si--Zhu in 2004 and offer also new families of mixed-mean inequalities.

math.CA↗

arXiv2019

Invariant means and iterates of mean-type mappings

Classical result states that for two continuous and strict means $M,\,N \colon I^2 \to I$ ($I$ is an interval) there exists a unique $(M,N)$-invariant mean $K \colon I^2 \to I$, i.e. such a mean that $K \circ (M,N)=K$ and, moreover, the sequence of iterates $((M,N)^n)_{n=1}^\infty$ converge to $(K,K)$ pointwise. Recently it was proved that continuity assumption cannot be omitted in general. We show that if $K$ is a unique $(M,N)$-invariant mean then, under no continuity assumption, $(M,N)^n \to (K,K)$.

math.CA↗

arXiv2018

Weakening of Hardy property for means

The aim of this paper is to find a broad family of means defined on a subinterval of $I \subset [0,+\infty)$ such that $$ \sum_{n=1}^\infty \mathscr{M}(a_1,\dots,a_n) <+\infty \quad\text{ for all }\quad a \in \ell_1(I).$$ Equivalently, the averaging operator $$ (a_1,\,a_2,a_3\,,\dots) \mapsto \big( a_1,\,\mathscr{M}(a_1,a_2),\,\mathscr{M}(a_1,a_2,a_3), \dots\big) $$ is a selfmapping of $\ell_1(I)$. This property is closely related to so-called Hardy inequality for means (which additionally requires boundedness of this operator). In fact we prove that these two properties are equivalent in a family of Gini means and Gaussian product of Power means. Moreover it is shown that this is not the case for quasi-arithmetic means.

math.CA↗

arXiv2018

On the homogenization of means

The aim of this paper is to introduce several notions of homogenization in various classes of weighted means, which include quasiarithmetic and semideviation means. In general, the homogenization is an operator which attaches a homogeneous mean to a given one. Our results show that, under some regularity or convexity assumptions, the homogenization of quasiarithmetic means are power means, and homogenization of semideviation means are homogeneous semideviation means. In other results, we characterize the comparison inequality, the Jensen concavity, and Minkowski- and Hölder-type inequalities related to semideviation means.

math.CA↗

arXiv2018

On a lattice-like property of quasi-arithmetic means

We will prove that in a family of quasi-arithmetic means sattisfying certain smoothness assumption (embed with a naural pointwise ordering) every finite family has both supremum and infimum, which is also a quasi-arithmetic mean sattisfying the same smoothness assumptions. More precisely, if $f$ and $g$ are $\mathcal{C}^2$ functions with nowhere vanishing first derivative then there exists a function $h$ such that: (i) $A^{[f]} \le A^{[h]}$, (ii) $A^{[g]} \le A^{[h]}$, and (iii) for every continuous strictly monotone function $s \colon I \to \mathbb{R}$ $$ A^{[f]} \le A^{[s]} \text{ and } A^{[g]} \le A^{[s]} \text{ implies } A^{[h]} \le A^{[s]} $$ ($A^{[f]}$ stands for a quasi-arithmetic mean generated by a function $f$ and so on). Moreover $h\in\mathcal{C}^2$, $h'\ne0$, and it is a solution of the differential equation $$ \frac{h''}{h'}=\max\Big(\frac{f''}{f'},\,\frac{g''}{g'}\Big). $$ We also provide some extension to a finite family of means. Obviously dual statements with inverses inequality sign as well as a multifuntion generalization will be also stated.

math.CA↗

arXiv2018

Invariant property for discontinuous mean-type mappings

It is known that if $M,\,N$ are continuous two-variable means such that $|M(x,y)-N(x,y)| < |x-y|$ for every $x,\ y$ with $x\ne y$, then there exists a unique invariant mean (which is continuous too). We are looking for invariant means for pairs satisfying the inequality above, but continuity of means is not assumed. In this setting the invariant mean is no longer uniquely defined, but we prove that there exist the smallest and the biggest one. Furthermore it is shown that there exists at most one continuous invariant mean related to each pair.

math.FA↗