Related papers: Bessel Potential Spaces and Complex Interpolation:…
We study (homogeneous and inhomogeneous) anisotropic Besov spaces associated to expansive dilation matrices $A \in {\rm GL}(d,\mathbb{R})$, with the goal of clarifying when two such matrices induce the same scale of Besov spaces. For this…
Indefinite inner product spaces of entire functions and functions analytic inside a disk are considered and their completeness studied. Spaces induced by the rotation invariant reproducing kernels in the form of the generalized…
The "potentials" being considered are analogues of classical Riesz potentials of order 1, and the idea is to look at how they might map L^p spaces into Sobolev spaces in various settings.
Within the new concept of a local iterated function system (local IFS), we consider a class of attractors of such IFSs, namely those that are graphs of functions. These new functions are called local fractal functions and they extend and…
We investigate the relationship between the compactness of embeddings of Sobolev spaces built upon rearrangement-invariant spaces into rearrangement-invariant spaces endowed with $d$-Ahlfors measures under certain restriction on the speed…
Classical convergence analysis for kernel interpolation typically assumes that the target function $f$ lies in the reproducing kernel Hilbert space $\mathcal{H}_k\!\left(\Omega\right)$ induced by a kernel on a domain…
We consider piecewise linear interpolation from the perspective of kernel interpolation and quadrature. If the Sobolev space $W_2^1(0, 1)$ is equipped with a suitable inner product, its reproducing kernel is piecewise linear and gives rise…
The notion of the ultrametrics can be considered as a zero-dimensional analogue of ordinary metrics, and it is expected to prove ultrametric versions of theorems on metric spaces. In this paper, we provide ultrametric versions of the…
In this paper, we study anisotropic Bessel potential and Besov spaces, where the anisotropy measures the extra amount of regularity in certain directions. Some basic properties of these spaces are established along with applications to…
Cocompactness is a property of embeddings between two Banach spaces, similar to but weaker than compactness, defined relative to some non-compact group of bijective isometries. In presence of a cocompact embedding, bounded sequences (in the…
In this article, via certain lower bound conditions on the measures under consideration, the authors fully characterize the Sobolev embeddings for the scales of Haj{\l}asz-Triebel-Lizorkin and Haj{\l}asz-Besov spaces in the general context…
In a recent paper, Yu. A. Brychkov derived a series of identities for multiples sums of special functions, using generating functions. Among these identities, a particularly interesting one involves multiples sums of Bessel $I_{\nu}$…
It is proved that the relation of isomorphism between separable Banach spaces is a complete analytic equivalence relation, i.e., that any analytic equivalence relation Borel reduces to it. Thus, separable Banach spaces up to isomorphism…
We prove uniqueness and stability for the inverse boundary value problem of the two dimensional Schr\"odinger equation. We do not assume the potentials to be continuous or even bounded. Instead, we assume that some of their positive…
Performing inference in Bayesian models requires sampling algorithms to draw samples from the posterior. This becomes prohibitively expensive as the size of data sets increase. Constructing approximations to the posterior which are cheap to…
In this paper we propose a unified approach, based on limiting interpolation, to investigate the embeddings for the Sobolev space $(\dot{W}^k_p(\mathcal{X}))_0, \, \mathcal{X} \in \{\mathbb{R}^d, \mathbb{T}^d, \Omega\}$, in the subcritical…
A number of new definite integrals involving Bessel functions are presented. These have been derived by finding new integral representations for the product of two Bessel functions of different order and argument in terms of the generalized…
The polynomial kernels are widely used in machine learning and they are one of the default choices to develop kernel-based classification and regression models. However, they are rarely used and considered in numerical analysis due to their…
An $RD$ space is a doubling measure metric space $\Omega$ with the additional property that it has a reverse doubling property. In this paper we introduce a new class of Haj{\l}asz-Besov spaces on $\Omega$ and extend several results from…
In this paper we prove Nikolskii's inequality on general compact Lie groups and on compact homogeneous spaces with the constant interpreted in terms of the eigenvalue counting function of the Laplacian on the space, giving the best constant…