A proposal to characterize and quantify superoscillations

Abstract

We present a formal definition of superoscillating function. We discuss the limitations of previously proposed definitions and illustrate that they do not cover the full gamut of superoscillatory behaviours. We demonstrate the suitability of the new proposal with several examples of well-known superoscillating functions that were not encompassed by previous definitions.

Figures (14)

• Figure 1: Comparison between the real part of $g(x, 2, 1000)$ and $\cos(2x)$, showing superoscillations around $x=0$.
• Figure 2: Comparison between $h(x, 1, 2)$ and a function proportional to $\sin(4x)$ in the interval $[0,\pi]$. The plot suggests that $h(x,1,2)$ is superoscillating around $x=\pi/2$.
• Figure 3: a) Comparison between the imaginary part of $g(x,2,10)$ and $\sin(2x)$ in the interval $[-2\pi,2\pi]$, showing that $\Im\{g(x,2,10)\}$ is superoscillating around $x=0$. b) Values of $Q_\text{sin}$ for intervals of the form $[0,b]$, as a function of the right endpoint $b>0$. The value of $Q_\text{sin}$ was plotted for consecutive zeros, until two points were reached for which $Q_\text{sin}<1/2$, indicating the end of the superoscillating interval.
• Figure 4: a) Comparison between the imaginary part of $g(x,2,20)$ and $\sin(2x)$ in the interval $[-2\pi,2\pi]$, showing that $\Im\{g(x,2,20)\}$ is superoscillating around $x=0$. b) Values of $Q_\text{sin}$ for intervals of the form $[0,b]$, as a function of the right endpoint $b>0$. The value of $Q_\text{sin}$ was plotted for consecutive zeros, until two points were reached for which $Q_\text{sin}<1/2$, indicating the end of the superoscillating interval.
• Figure 5: a) Comparison between the imaginary part of $g(x,2,20)+g(x,3,20)$ and $2\sin(5x/2)$ in the interval $[-\pi,\pi]$, showing that $\Im\{g(x,2,20)+g(x,3,20)\}$ is superoscillating around $x=0$. b) Values of $Q_\text{sin}$ for intervals of the form $[0,b]$, as a function of the right endpoint $b>0$. The value of $Q_\text{sin}$ was plotted for consecutive zeros, until two points were reached for which $Q_\text{sin}<1/2$, indicating the end of the superoscillating interval.

Theorems & Definitions (1)

• Definition : Superoscillating function in an interval