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

G. W. Wasilkowski

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

## Abstract

We study the error in approximating functions with a bounded (r + α)th derivative in an Lp-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.

Original language English 372-380 9 Journal of Approximation Theory 62 3 https://doi.org/10.1016/0021-9045(90)90059-Y Published - Sep 1990

## ASJC Scopus subject areas

• Analysis
• Numerical Analysis
• General Mathematics
• Applied Mathematics

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