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On the approximation of Weierstrass function via superoscillations

Fabrizio Colombo, Irene Sabadini, Daniele C. Struppa

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.

On the approximation of Weierstrass function via superoscillations

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.
Paper Structure (10 sections, 6 theorems, 80 equations, 1 figure)

This paper contains 10 sections, 6 theorems, 80 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Estimate of Truncated Weierstrass function with superoscillations
  3. Case 1: $y > 0$.
  4. Case 2: $-1 < y < 0$.
  5. Divergence of Fixed-$n$ Limit
  6. Case 1: $x = 0$.
  7. Case 2: $x \neq 0$.
  8. The Joint Convergence Theorem
  9. Interpretation of the Stability Limit
  10. Concluding Remarks

Key Result

Lemma 2.1

Let $\gamma > 0$ and $y > -1$. Then the following estimate holds:

Figures (1)

  • Figure 1: Stability phase diagram showing the Divergence Wall and the region of uniform convergence.

Theorems & Definitions (14)

  • Lemma 2.1
  • proof
  • Theorem 2.2: Explicit error estimate
  • proof
  • Definition 2.3: Truncated Weierstrass Function
  • Definition 2.4: Superoscillating Approximation
  • Theorem 2.5: Global Approximation Error
  • proof
  • Theorem 3.1: Divergence of the Limit $N \to \infty$
  • proof
  • ...and 4 more