Related papers: Simultaneous polynomial approximation in Beurling-…

Several local geometric properties of Orlicz space $L_\phi$ are presented for an increasing Orlicz function $\phi$ which is not necessarily convex, and thus $L_\phi$ does not need to be a Banach space. In addition to monotonicity of $\phi$…

In this paper, we deal with the family of Steklov sampling operators in the general setting of Orlicz spaces. The main result of the paper is a modular convergence theorem established following a density approach. To do this, a Luxemburg…

A broader class of Hardy spaces and Lebesgue spaces have been introduced recently on the unit circle by considering continuous $\|.\|_1$-dominating normalized gauge norms instead of the classical norms on measurable functions and a Beurling…

First we give here a simple proof of a remarkable result of Videnskii and Shirokov: let $B$ be a Blaschke product with $n$ zeros, then there exists an outer function $\phi, \phi(0)=1$, such that $\|(B\phi)'\| \leq C n$, where $C$ is an…

We prove an almost everywhere convergence result for Bochner-Riesz means of $L^p$ functions on Heisenberg-type groups, yielding the existence of a $p>2$ for which convergence holds for means of arbitrarily small order. The proof hinges on a…

Blaschke factorization allows us to write any holomorphic function $F$ as a formal series $$ F = a_0 B_0 + a_1 B_0 B_1 + a_2 B_0 B_1 B_2 + \cdots$$ where $a_i \in \mathbb{C}$ and $B_i$ is a Blaschke product. We introduce a more general…

We establish simultaneous approximation properties of weighted first-order Sobolev orthogonal projectors onto spaces of polynomials of bounded total degree in the Euclidean unit ball. The simultaneity is in the sense that we provide bounds…

Given Banach spaces E and F, we denote by ${\mathcal P}(^k!E,F)$ the space of all k-homogeneous (continuous) polynomials from E into F, and by ${\mathcal P}_{wb}(^k!E,F)$ the subspace of polynomials which are weak-to-norm continuous on…

We consider eigenfunctions of Schr\"odinger operators on a $d-$dimensional bounded domain $\Omega$ (or a $d-$dimensional compact manifold $\Omega$) with Dirichlet conditions. These operators give rise to a sequence of eigenfunctions…

The well-known Bohr--P\'al theorem asserts that for every continuous real-valued function $f$ on the circle $\mathbb T$ there exists a change of variable, i.e., a homeomorphism $h$ of $\mathbb T$ onto itself, such that the Fourier series of…

In this paper, we study complex analytic aspects of the moduli space $\Bcal_d^{fm}$ of degree $d\ge2$ fixed-point-marked Blaschke products. We define a complex structure on $\Bcal_d^{fm}$ and prove the simultaneous uniformization theorem…

We investigate affine Berkovich spaces over maximally complete fields and prove that they may be approximated by simpler spaces when the only functions we need to evaluate are polynomials of bounded degree. We derive applications to…

We investigate Sobolev spaces $W^{1,\Phi}$ associated to Musielak-Orlicz spaces $L^\Phi$. We first present conditions for the boundedness of the Voltera operator in $L^\Phi$. Employing this, we provide necessary and sufficient conditions…

We show that on separable Banach spaces admitting a separating polynomial, any uniformly continuous, bounded, real-valued function can be uniformly approximated by Lipschitz, analytic maps on bounded sets.

In this article, we investigate the existence of closed vector subspaces (i.e.spaceability) in various nonlinear subsets of Orlicz-Lorentz spaces $\Lambda_{\varphi,w}$, equipped with the Luxemburg norm. If a family of Orlicz functions…

By using Bernstein-type inequality we define analogs of spaces of entire functions of exponential type in $L_{p}(X), 1\leq p\leq \infty$, where $X$ is a symmetric space of non-compact. We give estimates of $L_{p}$-norms, $1\leq p\leq…

For $1 < r \le 2$, we study the set of monomial convergence for spaces of holomorphic functions over $\ell_r$. For $ H_b(\ell_r)$, the space of entire functions of bounded type in $\ell_r$, we prove that $\mbox{mon} H_b(\ell_r)$ is exactly…

The article deals with a simplified proof of the Sobolev embedding theorem for Lizorkin--Triebel spaces (that contain the $L_p$-Sobolev spaces $H^s_p$ as special cases). The method extends to a proof of the corresponding fact for general…

We show that in a scale of weighted Dirichlet spaces $D_{\alpha}$, including the Bergman space, given any finite Blaschke product $B$ there exists an equivalent norm in $D_{\alpha}$ such that $B$ satisfies the wandering subspace property…

Bernstein polynomials provide a constructive proof for the Weierstrass approximation theorem, which states that every continuous function on a closed bounded interval can be uniformly approximated by polynomials with arbitrary accuracy.…