On the approximation of Weierstrass function via superoscillations

TL;DR

This paper investigates how superoscillations can approximate the Weierstrass function, providing explicit error bounds and analyzing the convergence behavior of this approximation method.

Contribution

It offers the first detailed analysis of Berry's superoscillating approximation to the Weierstrass function, including explicit error estimates and convergence properties.

Findings

Explicit error bounds for superoscillating approximation

Analysis of convergence properties of double limits

Insights into the approximation of fractal functions

Abstract

The Weierstrass function is a classic example of a continuous nowhere differentiable function, defined as a sum of high-frequency complex exponentials. In this paper, we follow a suggestion of M.V. Berry and study the convergence properties of Berry's superoscillating approximation to the truncated Weierstrass function. We provide sharp, explicit error estimates for this approximation and we analyze the subtle convergence properties of the associated double limits.