On Multivariate Bernstein Polynomials

Abstract

In this paper, we first revisit and bring together as a sort of survey, properties of Bernstein polynomials of one variable. Secondly, we extend the results from one variable to several ones, namely—uniform convergence, uniform convergence of the derivatives, order of convergence, monotonicity, fixed sign for the p-th derivative, and deduction of the upper and lower bounds of Bernstein polynomials from those of the corresponding functions.

Cooperation between Independent Market Makers

“…The second quantity in the bracket only includes the situation that no agent selects the lowest spread and thus converges to 0. The first quantity converges to P(x * ), by the property of multivariate Bernstein polynomials and the continuity of P(•); see a recent survey in Foupouagnigni and Mouafo Wouodjié (2020).…”

Multivariate Zipper Fractal Functions

“…By Theorem 4.2, there exists a sequence {f α, ∆;n }, for n k ∈ N, of multivariate Bernstein zipper α-fractal functions that converges to f for any given non-negative function f ∈ C(I). By [22], B n is a positive linear operator and thus B n f (X) ≥ 0, for all X ∈ I, which implies the positivity of Φ n .…”

“…To get the convergence of multivariate α-fractal function f α, ∆,b to f without altering the scaling function α, we take as base functions b multivariate Bernstein polynomials B n f (X) [19,22] of f . The n := (n 1 , ..., n m )-th Bernstein polynomial for f ∈ C(I) is given by…”

“…These multivariate zipper fractal functions are more general than the existing classical and fractal approximants. Based on the existence of ZFIF, we construct a novel class of multivariate Bernstein zipper α-fractal functions using Bernstein polynomials B n1,...,nm f [19,22] as base functions in its IFS f α, n1n2...nm for a given f ∈ C m k=1 I k , where the I k are bounded and closed intervals in R. Multivariate Bernstein zipper α-fractal functions f α, n1n2...nm converge uniformly to f as n i → ∞ for all i, without altering the scaling functions. We prove that the multivariate Bernstein polynomial B n1,...,nm f is Lipschitz if so is f .…”

Doubly High-Dimensional Contextual Bandits: An Interpretable Model for Joint Assortment-Pricing

“…Furthermore, by suitably choosing the orders (N 1 , N 2 ) of lifting, such a function g can be used to approximate any continuous functions (a base , x base ) → f (a base , x base ) to arbitrary accuracy on any compact set C. (E.g., see the paper Foupouagnigni and Mouafo Wouodjié (2020) for results on the approximation (multivariate) functions with Bernstein polynomials). Note that besides polynomial type basis (EC.1) that yields the form (EC.2), we can also use other bases such as Fourier or Haar.…”