A New Constructive and Elementary Proof of a Bernstein–Walsh Theorem, Improved to Infinite Order Convergence, for Functions C∞ in an Intricate but Smooth Two-Dimensional Real Domain

Abstract

We give a constructive proof that a C∞ function f(x, y) can be approximated by a polynomial with an infinite order rate of convergence on a general two-dimensional domain that is specified as the set where a smooth function B(x, y) is non-negative and which can be embedded within a rectangle. We explicitly construct a C∞ smoothed approximation to the characteristic function of the domain as $\rho (x,y) \equiv \mathcal {H}([1 + SB(x,y)])$ where S > 0 is a constant, $\mathcal {H}$ is a “ramp” (a smoothed approximation to the Heaviside step function) and ρ ≡ 1 on the domain Ω. The product f(x, y)ρ(x, y) is identically equal to f on the domain, but is of compact support. From this, we prove that ρf has a bivariate Chebyshev series on a rectangle that embeds Ω. We prove also that this expansion converges with increasing N, where N is the series truncation, faster than any finite inverse power of N, which is the definition of “infinite order” convergence. Bernstein–Walsh-type theorems have a long history, but the proofs use mathematical tools far removed from the education of the engineers and scientists. In contrast, our proof is constructive using tools accessible to applied practitioners. The proof of exponential convergence for functions with only C∞ smoothness has not been previously given by any methodology.