A Note on Bernstein Polynomial Type Approximations

On Bernstein Type Inequalities for Complex Polynomial

In this paper, we establish some Bernstein type inequalities for the complex polynomial. Our results constitute generalizations and refinements of some well-known polynomial inequalities.

Iterated Bernstein polynomial approximations

Iterated Bernstein polynomial approximations of degree n for continuous function which also use the values of the function at i/n, i = 0, 1, . . . , n, are proposed. The rate of convergence of the classic Bernstein polynomial approximations is significantly improved by the iterated Bernstein polynomial approximations without increasing the degree of the polynomials. The same idea applies to the...

A note on Bernstein and Markov type inequalities

Let D be the unit disk in the complex plane C . We prove that for any polynomial p of degree at most n max z ∈ D ∣∣∣p(z) − p(z̄) z − z̄ ∣∣∣ n max 0 j n ∣∣∣∣∣ p ( eij /n ) + p ( e−ij /n )

A note on computational complexities of polynomial type

We introduce a notion of a polynomial type complexity of a multivariate polynomial. We provide examples of polynomial type complexities, including the determinantal complexity. We present two algorithms (not very efficient) to construct test functions for lower bounds (or explicit obstructions) of a polynomial type complexity, one of them is based on Gröbner bases, the other uses the resultant ...

Note on a polynomial of