On piecewise-polynomial approximation of functions with a bounded fractional derivative in an L_p-norm

We study the error in approximating functions with a bounded (r + α)th derivative in an L_p-norm. Here r is a nonnegative integer, α ε{lunate} [0, 1), and f{hook}^{(r + α)} is the classical fractional derivative, i.e., f{hook}^{(r + α)}(y) = ∝₀¹, ^α d(f{hook}^(r)(t)). We prove that, for any such function f{hook}, there exists a piecewise-polynomial of degree s that interpolates f{hook} at n equally spaced points and that approximates f{hook} with an error (in sup-norm) ∥f{hook}^{(r + α)}∥_p O(n^{-(r+α- 1 p}). We also prove that no algorithm based on n function and/or derivative values of f{hook} has the error equal ∥f{hook}^{(r + α)}∥_p O(n^{-(r+α- 1 p}) for any f{hook}. This implies the optimality of piecewise-polynomial interpolation. These two results generalize well-known results on approximating functions with bounded rth derivative (α = 0). We stress that the piecewise-polynomial approximation does not depend on α nor on p. It does not depend on the exact value of r as well; what matters is an upper bound s on r, s ≥ r. Hence, even without knowing the actual regularity (r, α, and p) of f{hook}, we can approximate the function f{hook} with an error equal (modulo a constant) to the minimal worst case error when the regularity were known.