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A Banach space is said to have the Lebesgue property if every Riemann-integrable function $f:[0,1]\to X$ is Lebesgue almost everywhere continuous. We give a characterization of the Lebesgue property in terms of a new sequential asymptotic…
We derive in this preprint the exact up to multiplicative constant non-asymptotical estimates for the norms of some non-linear in general case operators, for example, the so-called maximal functional operators, in two probabilistic…
In this paper, we show that Besov and Triebel-Lizorkin functions can be approximated by a H\"older continuous function both in the Lusin sense and in norm. The results are proven in metric measure spaces for Haj{\l}asz-Besov and…
We combine functional analytic and geometric viewpoints on approximate Birkhoff and isosceles orthogonality in generalized Minkowski spaces which are finite-dimensional vector spaces equipped with a gauge. This is the first approach to…
We apply the metrical approach to Sobolev spaces, which arise in various evolution PDEs. Functions from those spaces are defined on an interval and take values in a family of Banach spaces. In this case we adapt the definition of Newtonian…
We consider the task of computing an approximate minimizer of the sum of a smooth and non-smooth convex functional, respectively, in Banach space. Motivated by the classical forward-backward splitting method for the subgradients in Hilbert…
This paper investigates the approximation of Gaussian random variables in Banach spaces, focusing on the high-probability bounds for the approximation of Gaussian random variables using finitely many observations. We derive non-asymptotic…
Starting with an integrable unitary representation of a locally compact group and its associated voice transform, coorbit theory describes the construction and investigation of the so-called coorbit spaces. A coorbit space consists of…
Using partial derivatives $\partial_zf$ and $\partial_{\ol z}f$, we introduce Besov spaces of polyanalytic functions on the unit disk and on the upper half-plane. We then prove that the dilatations of each function in polyanalytic Besov…
We develop local elliptic regularity for operators having coefficients in a range of Sobolev-type function spaces (Bessel potential, Sobolev-Slobodeckij, Triebel-Lizorkin, Besov) where the coefficients have a regularity structure typical of…
We introduce a Banach space-valued extension of random feature learning, a data-driven supervised machine learning technique for large-scale kernel approximation. By randomly initializing the feature maps, only the linear readout needs to…
We introduce Besov and Triebel--Lizorkin spaces on a manifold with boundary adapted to H\"ormander vector fields, near a so-called non-characteristic point of the boundary. We prove sharp results in these spaces for the corresponding…
In the article, integration of temporal functions in (possibly non-UMD) Banach spaces with respect to (possibly non-Gaussian) fractional processes from a finite sum of Wiener chaoses is treated. The family of fractional processes that is…
We characterize the relatively compact subsets of $L^1\left(\| m \| \right),$ the quasi-Banach function space associated to the semivariation of a given vector measure $m$ showing that the strong connection between compactness, uniform…
Low-rank tensor approximation techniques attempt to mitigate the overwhelming complexity of linear algebra tasks arising from high-dimensional applications. In this work, we study the low-rank approximability of solutions to linear systems…
In this paper we prove structural and topological characterizations of the screened Sobolev spaces with screening functions bounded below and above by positive constants. We generalize a method of interpolation to the case of seminormed…
In this paper, we consider sampling in a reproducing kernel subspace of $L^p$. We introduce a pre-reconstruction operator associated with a sampling scheme and propose a Galerkin reconstruction in general Banach space setting. We show that…
This paper provides a self-contained exposition of coorbit spaces associated to integrable group representations and quasi-Banach function spaces, and at the same time extends and simplifies previous work. The main results provide an…
In recent years, the ergodic theory of group actions on homogeneous spaces has played a significant role in the metric theory of Diophantine approximation. We survey some recent developments with special emphasis on Diophantine properties…
In this paper we obtain the non-asymptotic norm estimations of Besov's type between the norms of a functions in different Bilateral Grand Lebesgue spaces (BGLS). We also give some examples to show the sharpness of these inequalities.