[Paper Review] On convergence of partial derivatives of multivariate Bernstein polynomials

A. Yu. Veretennikov, Evguenia Veretennikova|arXiv (Cornell University)|Jul 18, 2015

Approximation Theory and Sequence Spaces2 references3 citations

TL;DR

This paper establishes that multivariate Bernstein polynomials converge to both a function and its partial derivatives when the derivatives are continuous, extending classical univariate results to higher dimensions. The proof leverages finite difference approximations and the multivariate law of large numbers, providing a rigorous foundation for applications in stochastic calculus, particularly in deriving multi-dimensional Itô's formula.

ABSTRACT

It is be shown that the sequence of Bernstein polynomials for a function of several variables converges to this function uniformly along with every partial derivative of any order, provided that the latter derivative is well defined and continuous. This may be interesting in some questions of stochastic calculus, in particular, in a methodical proof of multidimensional Ito's formula based on the product rule.

Motivation & Objective

• To establish the convergence of partial derivatives of multivariate Bernstein polynomials to the true partial derivatives of a function.
• To extend the classical univariate Bernstein polynomial convergence result—valid for functions and their derivatives—to the multivariate case.
• To provide a rigorous proof for the approximation of second-order derivatives in the context of multi-dimensional Itô's formula.
• To address the lack of explicit references in the literature on derivative convergence for multivariate Bernstein polynomials.

Proposed method

• Derives a multivariate generalization of the univariate Bernstein polynomial derivative formula using multi-index notation and finite differences.
• Employs the identity that the k-th order partial derivative of the Bernstein polynomial is expressed as a weighted sum of k-th order finite differences of the function.
• Uses the normalized finite difference operator Δ^k_f(x) = n^|k| Δ_{1/n}^k f(x), which converges uniformly to the k-th order partial derivative as n → ∞.
• Applies the multivariate law of large numbers to show that the Bernstein polynomial derivatives converge uniformly on compact sets.
• Establishes the convergence via induction on the multi-index k, proving the derivative formula holds for all orders of differentiation.
• Relies on combinatorial identities involving multinomial coefficients and reindexing to derive the recursive structure of the derivative expressions.

Experimental results

Research questions

• RQ1 Does the multivariate Bernstein polynomial approximation preserve convergence of partial derivatives when the derivatives are continuous?
• RQ2 Can the classical univariate result on derivative convergence of Bernstein polynomials be generalized to functions of several variables?
• RQ3 Is the convergence of partial derivatives of multivariate Bernstein polynomials uniform on compact subsets of the domain?
• RQ4 What is the role of finite difference approximations in the convergence of multivariate Bernstein polynomial derivatives?
• RQ5 How does the multivariate Bernstein operator preserve derivative information, and why is this important for stochastic calculus?

Key findings

• Multivariate Bernstein polynomials converge uniformly to the original function and all its partial derivatives of order up to k, provided the derivatives are continuous.
• The k-th order partial derivative of the Bernstein polynomial is expressed as a sum involving k-th order finite differences of the function, scaled by multinomial coefficients.
• The normalized finite difference Δ_{1/n}^k f(x) converges uniformly to the k-th order partial derivative ∂^k f / ∂x^k as n → ∞.
• The convergence of the polynomial derivatives is uniform on compact subsets of the standard simplex in ℝ^d.
• The result provides a constructive and probabilistic justification for the use of Bernstein polynomials in proving multi-dimensional Itô's formula.
• The proof technique extends the univariate approach via the law of large numbers to the multivariate setting using multinomial distributions.