Pointwise bounded approximation by polynomials

Uniform and Pointwise Shape Preserving Approximation by Algebraic Polynomials

We survey developments, over the last thirty years, in the theory of Shape Preserving Approximation (SPA) by algebraic polynomials on a finite interval. In this article, “shape” refers to (finitely many changes of) monotonicity, convexity, or q-monotonicity of a function. It is rather well known that it is possible to approximate a function by algebraic polynomials that preserve its shape (i.e....

Bounded Approximation by Polynomials Whose Zeros Lie on a Circleo

A Pointwise Approximation Theorem for Linear Combinations of Bernstein Polynomials

Recently, Z. Ditzian gave an interesting direct estimate for Bernstein polynomials. In this paper we give direct and inverse results of this type for linear combinations of Bernstein polynomials.

Pointwise Approximation by Elementary Complete Contractions

A complete contraction on a C-algebra A, which preserves all closed two sided ideals J , can be approximated pointwise by elementary complete contractions if and only if the induced map on B⊗A/J is contractive for every C-algebra B, ideal J in A and C-tensor norm on B ⊗ A/J . A lifting obstruction for such an approximation is also obtained.

Approximation by homogeneous polynomials

A new, elementary proof is given for the fact that on a centrally symmetric convex curve on the plane every continuous even function can be uniformly approximated by homogeneous polynomials. The theorem has been proven before by Benko and Kroó, and independently by Varjú using the theory of weighted potentials. In higher dimension the new method recaptures a theorem of Kroó and Szabados, which ...