The Journal of Brief Ideas

Interpolation and approximation of continuous functions by Hermite polynomials

The Chebyshev points in [-1,1] are the zeros of the Chebyshev polynomials.

Fejér uniformly approximated arbitrary continuous functions f:[-1,1]→ ℝ by use of Hermite polynomials at Chebyshev points (Interpolation and Approximation, P. J. Davis, 1963, pp. 118-121). These are polynomials that agree with the function at the Chebyshev points and whose derivative there is 0.

Chebyshev points have a β(1/2, 1/2) distribution. The generic orbits of the chaotic function 4x(1-x) also have a β(1/2, 1/2) distribution (1), and a large enough finite portion of such an orbit lies very close to the the Chebyshev points: could we use these points in place of Chebyshev points? More generally:

Question: If f:[-1,1] → ℝ is a continuous function and for each integer n, Γ(n) is a random sample of n points from a β (1/2,1/2) distribution on [-1,1] do the Hermite-Fejér polynomials corresponding to the grid points Γ(n) converge uniformly to f with probability 1?

The answer is likely "yes": "Polynomial Approximation" by Gordon K. Smyth, pp. 5-6.

Can we formulate other ideas of numerical analysis as statistical theorems (with convergence in probability)? (2).