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Using notions from the geometry of Banach spaces we introduce square functions $\gamma(\Omega,X)$ for functions with values in an arbitrary Banach space $X$. We show that they have very convenient function space properties comparable to the…

We study the space spanned by the integer shifts of a bivariate Gaussian function and the problem of reconstructing any function in that space from samples scattered across the plane. We identify a large class of lattices, or more generally…

In this paper, we consider a Banach space valued random coefficient autoregressive process. Our studies on this process involve existence, weak law of large numbers, strong law of large numbers, some exponential inequalities, central limit…

The concept of bounded variation has been generalized in many ways. In the frame of functions taking values in Banach space, the concept of bounded semivariation is a very important generalization. The aim of this paper is to provide an…

Given any Banach space $X$, let $L_2^X$ denote the Banach space of all measurable functions $f:[0,1]\to X$ for which ||f||_2:=(int_0^1 ||f(t)||^2 dt)^{1/2} is finite. We show that $X$ is a UMD--space (see \cite{BUR:1986}) if and only if…

We consider estimating the shared mean of a sequence of heavy-tailed random variables taking values in a Banach space. In particular, we revisit and extend a simple truncation-based mean estimator first proposed by Catoni and Giulini. While…

This work explores the equivalence of two sequential properties, $D$ and $D'$, for dual Banach spaces under the weak* topology. Property $D$ ensures that any totally scalarly measurable function is also scalarly measurable, while property…

In this paper we introduce a class of Banach spaces of functions of class C^r (where r is a positive real number) and the associated dual spaces of distributions of order r, which turn out to be useful in p-adic Langlands theory. We…

We prove large and moderate deviation results for sequences of compound sums, where the summands are i.i.d. random variables taking values in a separable Banach space. We establish that the results hold by proving that we are dealing with…

The spatial distribution has been widely used to develop various nonparametric procedures for finite dimensional multivariate data. In this paper, we investigate the concept of spatial distribution for data in infinite dimensional Banach…

In many textbooks on Generalized Functions and Distributions, we can find an example of infinitely differentiable function of bounded support from space D, called a bump function. This example is incorrect since this function is not…

We present several results providing lower bounds for the Banach-Mazur distance \[d_{BM}(C(K), C(L))\] between Banach spaces of continuous functions on compact spaces. The main focus is on the case where $C(L)$ represents the classical…

We describe the spaces $H^1(R)$ and BMO$(R)$ in terms of their closely related, simpler dyadic and two-sided counterparts. As a result of these characterizations we establish when a bounded linear operator defined on dyadic or two-sided…

In this manuscript we consider random objects being measured in multiple metric spaces, which may arise when those objects may be measured in multiple distinct ways. In this new multivariate setting, we define a Fr\'echet covariance and…

In this work we provide a characterization of distinct type of (linear and non-linear) maps between Banach spaces in terms of the differentiability of certain class of Lipschitz functions. Our results are stated in an abstract bornological…

We investigate the distance function $\boldsymbol{\delta}_{K}^{\phi}$ from an arbitrary closed subset $ K $ of a~finite-dimensional Banach space $ (\mathbf{R}^{n}, \phi) $, equipped with a uniformly convex $\mathcal{C}^{2}$-norm $ \phi $.…

Newtonian spaces generalize first-order Sobolev spaces to abstract metric measure spaces. In this paper, we study regularity of Newtonian functions based on quasi-Banach function lattices. Their (weak) quasi-continuity is established,…

We establish a general criterion for the existence of convex sets of fixed shape as, e.g., balls of a given radius, of maximal probability on Banach spaces. We also provide counterexamples showing that their existence my fail even in some…

In [2] we characterized in terms of a quadratic growth condition various metric regularity properties of the subdifferential of a lower semicontinuous convex function acting in a Hilbert space. Motivated by some recent results in [16] where…

Given a positive operator-valued measure $\nu$ acting on the Borel sets of a locally compact Hausdorff space $X$, with outcomes in the algebra $\mathcal B(\mathcal H)$ of all bounded operators on a (possibly infinite-dimensional) Hilbert…