Problems requiring analysis in higher-dimensional spaces have appeared naturally in electrical engineering, computer science, mathematics, physics, and statistics. In many cases, these problems focus on objects determined by an infinite number of parameters and/or are defined by functions of an infinite number of variables. They are currently studied using analytic, combinatorial, geometric and probabilistic methods from functional analysis. This paper is devoted to one of the important missing tools, a reasonable (or constructive) theory of Lebesgue measure for separable Banach spaces. A reasonable theory is one that provides: (1) a direct constructive extension of the finite-dimensional theory; and, (2) most (if not all) of the analytic tools available in finite dimensions. We approach this problem by embedding every separable Banach space into \(\mathbb{R}^\infty\) and use the unique \(\sigma {\text{-finite}}\) Lebesgue measure defined on this space as a bridge to the construction of a Lebesgue integral on every separable Banach space as a limit of finite-dimensional integrals. In our first application we define universal versions of Gaussian and Cauchy measure for every separable Banach space, which are absolutely continuous with respect to our Lebesgue measure. As our second application we constructively solve the diffusion equation in infinitely-many variables and introduce the interesting climate model problem of P. D. Thompson defined on infinite-dimensional phase space.

We survey several properties of the class of strictly singular operators defined on Banach lattices of measurable functions and rearrangement invariant function spaces. The related classes of disjointly strictly singular operators and super strictly singular operators are also discussed. In particular we present interpolation properties of strictly singular operators between \(L_p-L_q\) spaces.

We discuss recent results and open problems connected with distributions of polynomials in many or infinitely many variables on spaces with Gaussian measures and Nikolskii--Besov spaces of fractional smoothness related to such distributions.

These problems are considered for a wide range of orthogonal series. That is to say for trigonometric, Walsh, and other series on systems of characters of zero-dimensional compact abelian groups with the second axiom of countability. By this the sets of uniqueness are regarded for one-dimensional and multiple orthogonal series for many types of convergence.

Consider the action of \(SL(n+1,\mathbb{R})\) on \(\mathbb{S}^n\) arising as the quotient of the linear action on \(\mathbb{R}^{n+1}\setminus\{0\}\). We show that for a semigroup \(\mathfrak{S}\) of \(SL(n+1,\mathbb{R})\), the following are equivalent: \((1)\) \(\mathfrak{S}\) acts distally on the unit sphere \(\mathbb{S}^n\). \((2)\) the closure of \(\mathfrak{S}\) is a compact group. We also show that if \(\mathfrak{S}\) is closed, the above conditions are equivalent to the condition that every cyclic subsemigroup of \(\mathfrak{S}\) acts distally on \(\mathbb{S}^n\). On the unit circle \(\mathbb{S}^1\), we consider the ‘affine’ actions corresponding to maps in \(GL(2,\mathbb{R})\) and discuss the conditions for the existence of fixed points and periodic points, which in turn imply that these maps are not distal.

The discrete version of Milne’s inequality and its converse states that \begin{equation*} (*)\quad \sum_{j=1}^n\frac{w_j}{1-p_j^2} \leq \sum_{j=1}^n\frac{w_j}{1-p_j} \sum_{j=1}^n\frac{w_j}{1+p_j} \leq \Bigl(\sum_{j=1}^n\frac{w_j}{1-p_j^2} \Bigr)^2 \end{equation*} is valid for all \(w_j>0\) \((j=1,...,n)\) with \(w_1+\dots+w_n=1\) and \(p_j\in (-1,1)\) \((j=1,...,n)\). We present new upper and lower bounds for the product \(\sum w/(1-p) \sum w/(1+p)\). In particular, we obtain an improvement of the right-hand side of \((*)\). Moreover, we prove a matrix analogue of our double-inequality.

In this paper, we introduce and study a certain type of sub-semigroup of \(\mathbb{R}/\mathbb{Z}\) which turns out to be closely related to Szemerédi’s theorem on arithmetic progressions.

Let \(\mathcal{E}(X,d,\mu)\) be a Banach function space over a space of homogeneous type \((X,d,\mu)\). We show that if the Hardy-Littlewood maximal operator \(M\) is bounded on the space \(\mathcal{E}(X,d,\mu)\), then its boundedness on the associate space \(\mathcal{E}'(X,d,\mu)\) is equivalent to a certain condition \(\mathcal{A}_\infty\). This result extends a theorem by Andrei Lerner from the Euclidean setting of \(\mathbb{R}^n\) to the setting of spaces of homogeneous type.

The purpose of this paper is to give some characterizations of the weight functions \(w\) such that \(Mw\in A_{\infty }\left( \mathbb{R}^{n}\right) \). We show that, for these \(Mw\) weights, being in \(A_{\infty }\) ensures being in \(A_{1}\). We give a criterion in terms of the local maximal functions \(% m_{\lambda }\) and we present a pair of applications, one of them similar to the Coifman-Rochberg characterization of \(A_{1}\) but using functions of the form \(\left( f^{\#}\right) ^{\delta }\) and \(\left( m_{\lambda }u\right) ^{\delta }\) instead of \(\left( Mf\right) ^{\delta }\).

For a compact set \(K\subseteq\mathbb{R}^m\), we have two indexes given under simple parameters of the set \(K\) (these parameters go back to Besicovitch and Taylor in the late 1950’s). In the present paper we prove that with the exception of a single extreme value for each index, we have the following elementary estimate on how fast the ratio in the strong density theorem of Saks will tend to one: \[ \frac{|R\cap K|}{|R|}>1-o\bigg(\frac{1}{|\log d(R)|}\bigg) \qquad \text{for a.e.} \ \ x\in K \ \ \text{and for} \ \ d(R)\to0 \] (provided \(x\in R\), where \(R\) is an interval in \(\mathbb{R}^m\), \(d\) stands for the diameter, and \(|\cdot|\) is the Lebesgue measure).

This work is a natural sequence of [3] and constitutes a contribution to Problem 146 of Ulam [5, p. 245] (see also [8, p.78]) and Erdös' Scottish Book ‘Problems’ [5, Chapter 4, pp. 27-33], since it is known that no general statement can be made on how fast the density will tend to one.

We introduce the notion of the the weak integral by partitions of unity for functions defined on a \(\sigma\)-finite outer regular quasi Radon measure space \((S,\Sigma,\mathcal{T},\mu)\) into a Banach space \(X\) and discuss its relation with the weak McShane integral which has been introduced by M. Saadoune and R. Sayyad (2014).

A central question in the field of inhomogeneous attractors has been to relate the dimension of an inhomogeneous attractor to the condensation set and associated homogeneous attractor. This has been achieved only in specific settings, with notable results by Olsen, Snigireva, Fraser and Käenmäki on inhomogeneous self-similar sets, and by Burrell and Fraser on inhomogeneous self-affine sets. This paper is devoted to filling a significant gap in the dimension theory of inhomogeneous attractors, by studying those formed from arbitrary bi-Lipschitz contractions. We show that the maximum of the dimension of the condensation set and a quantity related to pressure, which we term upper Lipschitz dimension, forms a natural and general upper bound on the dimension. Additionally, we begin a new line of enquiry; the methods developed are used to classify the Hausdorff measure of inhomogeneous attractors. Our results have applications for affine systems with affinity dimension less than or equal to one and systems satisfying bounded distortion, such as conformal systems in dimensions greater than one.

It is shown how the Steinhaus property and ergodicity of a translation invariant extension \(\mu\) of the Lebesgue measure depend on the measure-theoretic density of \(\mu\)-measurable sets. Some connection of the Steinhaus property with almost convex sets is considered and a translation invariant extension of the Lebesgue measure is presented, for which the generalized Steinhaus property together with the mid-point convexity do not imply the almost convexity.

In the present paper we recall the main points of the Fourier Transform developments, in particular the historical origin of the inversion formula. Hence we construct explicit examples of functions in different zones of the range of the Fourier transform in \(L^1\). These can be used as exercises in a basic course of signal processing or harmonic analysis.