Necessary and sufficient conditions are given for the reflected Cauchy's operator (the reflected double layer potential operator) to be continuous as an operator from the space of all continuous functions on the boundary of the investigated domain to the space of all holomorphic functions on this domain (to the space of all harmonic functions on this domain) equipped with the topology of locally uniform convergence.

We introduce and study a new class of locally convex vector lattices of continuous functions on a locally compact Hausdorff space, which we call regular vector lattices. We investigate some general properties of these spaces and of the subspaces of so-called generalized affine functions. Moreover, we present some Korovkin-type theorems for continuous positive linear operators; in particular, we study Korovkin subspaces for finitely defined operators, for the identity operator and for positive...

We develop a theory of removable singularities for the weighted Bergman space ${𝒜}_{\mu }^p\left(\Omega \right)=\{f\text{analytic}\text{in}\Omega {\int }_{\Omega }{\left|f\right|}^p\mathrm{d}\mu <\infty \}$, where $\mu$ is a Radon measure on $ℂ$. The set $A$ is weakly removable for ${𝒜}_{\mu }^p(\Omega \setminus A)$ if ${𝒜}_{\mu }^p(\Omega \setminus A)\subset \text{Hol}\left(\Omega \right)$, and strongly removable for ${𝒜}_{\mu }^p(\Omega \setminus A)$ if ${𝒜}_{\mu }^p(\Omega \setminus A)={𝒜}_{\mu }^p\left(\Omega \right)$. The general theory developed is in many ways similar to the theory of removable singularities for Hardy $H^p$ spaces, $\mathrm{B}MO$ and locally Lipschitz spaces of analytic functions, including the existence of counterexamples to many plausible properties, e.g. the union of two compact removable singularities needs not be removable....

We give a representation of the spaces $C^{\infty }\left({ℝ}^N\right)\cap H^{k,p}\left({ℝ}^N\right)$ as spaces of vector-valued sequences and use it to investigate their topological properties and isomorphic classification. In particular, it is proved that $C^{\infty }\left({ℝ}^N\right)\cap H^{k,2}\left({ℝ}^N\right)$ is isomorphic to the sequence space $s^{\mathbb{N}}\cap l^2\left(l^2\right)$, thereby showing that the isomorphy class does not depend on the dimension N if p=2.