Pointwise convergence problem of free Benjamin-Ono-Burgers equation

<abstract><p>In this paper, we show the almost everywhere pointwise convergence of free Benjamin-Ono-Burgers equation in $ H^{s}({\bf{R}}) $ with $ s &gt; 0 $ with the aid of the maximal function estimate.</p></abstract>

Real Functions, Covers and Bornologies

The paper tries to survey the recent results about relationships between covering properties of a topological space X and the space USC(X) of upper semicontinuous functions on X with the topology of pointwise convergence. Dealing with properties of continuous functions C(X), we need shrinkable covers. The results are extended for A-measurable and upper A-semimeasurable functions where A is a family of subsets of X. Similar results for covers respecting a bornology and spaces USC(X) or C(X) endowed by a topology defined by using the bornology are presented. Some of them seem to be new.

Around Taylor’s Theorem on the Convergence of Sequences of Functions

Egoroff’s classical theorem shows that from a pointwise convergence we can get a uniform convergence outside the set of an arbitrary small measure. Taylor’s theorem shows the possibility of controlling the convergence of the sequences of functions on the set of the full measure. Namely, for every sequence of real-valued measurable factions |fn } n∈ℕ pointwise converging to a function f on a measurable set E, there exist a decreasing sequence |δn } n∈ℕ of positive reals converging to 0 and a set A ⊆ E such that E \ A is a nullset and lim n → + ∞ | f n ( x ) − f ( x ) | δ n = 0 for all x ∈ A . Let J ( A , { f n } ) {\lim _{n \to + \infty }}\frac{{|{f_n}(x) - f(x)|}}{{{\delta _n}}} = 0\,{\rm{for}}\,{\rm{all}}\,x \in A.\,{\rm{Let}}\,J(A,\,\{ {f_n}\} ) denote the set of all such sequences |δn } n∈ℕ. The main results of the paper concern basic properties of sets of all such sequences for a given set A and a given sequence of functions. A relationship between pointwise convergence, uniform convergence and the Taylor’s type of convergence is considered.

ON POINTWISE APPROXIMATION OF FUNCTIONS OF SEVERAL VARIABLES BY SINGULAR INTEGRALS

We prove a theorem on weighted pointwise convergence of \ multidimensional integral operators with radial kernels to generating function of several variables, which are in general non-integrable in $n$-dimensional Euclidean space $E_{n}$ in the sense of Lebesgue$.$ Main result holds at almost every point of $E_{n}.$

A Class of Integral Operators that Fix Exponential Functions

In this paper we introduce a general class of integral operators that fix exponential functions, containing several recent modified operators of Gauss–Weierstrass, or Picard or moment type operators. Pointwise convergence theorems are studied, using a Korovkin-type theorem and a Voronovskaja-type formula is obtained.

Convex-Cyclic Weighted Composition Operators and Their Adjoints

We characterize the convex-cyclic weighted composition operators $$W_{(u,\psi )}$$ W ( u , ψ ) and their adjoints on the Fock space in terms of the derivative powers of $$ \psi $$ ψ and the location of the eigenvalues of the operators on the complex plane. Such a description is also equivalent to identifying the operators or their adjoints for which their invariant closed convex sets are all invariant subspaces. We further show that the space supports no supercyclic weighted composition operators with respect to the pointwise convergence topology and, hence, with the weak and strong topologies, and answers a question raised by T. Carrol and C. Gilmore in [5].

On linear continuous operators on C_p-spaces

The paper describes the structure of a linear continuous operator on the space of continuous functions in the topology of pointwise convergence. The corresponding theorem is a generalization of A.V.Arkhangel'skii's theorem on the general form of a continuous linear functional on such spaces.