Résumé. Soient D un ouvert de ℂ et E un compact de D. Moyennant une hypothèse assez faible sur D et ℂ̅ E on montre que si α ∈ ]0,1[ vérifie $\partial D_{\alpha }\subset DE$, $D_{\alpha }$ étant l’ouvert de niveau z ∈ D : ω(E,D,z) < α, alors toute base commune de O(E) et O(D) est une base de $O\left(D_{\alpha }\right)$.

Some basic theorems and formulae (equations and inequalities) of several areas of mathematics that hold in Bernstein spaces $B_{\sigma }^p$ are no longer valid in larger spaces. However, when a function f is in some sense close to a Bernstein space, then the corresponding relation holds with a remainder or error term. This paper presents a new, unified approach to these errors in terms of the distance of f from $B_{\sigma }^p$. The difficult situation of derivative-free error estimates is also covered.

Associated with some properties of weighted composition operators on the spaces of bounded harmonic and analytic functions on the open unit disk $$𝔻$$ , we obtain conditions in terms of behavior of weight functions and analytic self-maps on the interior $$𝔻$$ and on the boundary $$\partial 𝔻$$ respectively. We give direct proofs of the equivalence of these interior and boundary conditions. Furthermore we give another proof of the estimate for the essential norm of the difference of weighted composition operators.

Given 0 < p,q < ∞ and any sequence z = zₙ in the unit disc , we define an operator from functions on to sequences by $T_{z,p}\left(f\right)={(1-|zₙ\left|²\right)}^{1/p}f\left(zₙ\right)$. Necessary and sufficient conditions on zₙ are given such that $T_{z,p}$ maps the Hardy space $H^p$ boundedly into the sequence space ${\ell }^q$. A corresponding result for Bergman spaces is also stated.