The various Artin approximation theorems assert the existence of power series solutions of a certain quality Q : formal, analytic, algebraic, of systems of equations of the same quality Q, assuming the existence of power series solutions of a weaker quality Q’ < Q : approximated, formal. The results are frequently used in commutative algebra and algebraic geometry. We present a systematic argum...

The aim of the paper is to prove that if L is a linear subspace of the space C(K) of all real-valued continuous functions defined on a nonempty compact Hausdorff space K such that min(|f |, 1) ∈ L whenever f ∈ L, then for any nonzero g ∈ L̄ (where L̄ denotes the uniform closure of L in C(K)) and for any sequence (bn)n=1 of positive numbers satisfying the relation P∞ n=1 bn = ‖g‖ there exists a se...

A type-2 computable real function is necessarily continuous; and this remains true for computations relative to any oracle. Conversely, by the Weierstrass Approximation Theorem, every continuous f : [0; 1]→ R is computable relative to some oracle. In their search for a similar topological characterization of relatively computable multi-valued functions f : [0; 1] ⇒ R (also known as multi-functi...

We prove the following theorem about Julia sets of the maps fn,p,γ(z) = z n + γ zp , for integers n, p ≥ 2, γ ∈ C by using techniques developed for the Weierstrass elliptic ℘ function and adapted to this setting. Folklore connectedness theorem: If fn,p,γ has a bounded critical orbit, then J(fn,p,γ) is connected. This is related to connectivity results by the author and others about J(℘), where ...

An alternative approach, known today as the Bernstein polynomials, to Weierstrass uniform approximation theorem was provided by Bernstein. These basis polynomials have attained increasing momentum, especially in operator theory, integral equations and computer-aided geometric design. Motivated improvements of computational disciplines, we propose a new generalization Bernstein–Kantorovich opera...

We show that the Weierstrass points of the generic curve of genus g over an algebraically closed field of characteristic 0 generate a group of maximal rank in the Jacobian. The Weierstrass points are a set of distinguished points on curves, which are geometrically intrinsic. In particular, the group these points generate in the Jacobian is a geometric invariant of the curve. A natural question ...