Beyond Superoscillation: General Theory of Approximation with Bandlimited Functions

TL;DR

This paper develops a general theoretical framework for constructing bandlimited functions that exhibit superoscillation and supergrowth, enabling precise control over their behavior using polynomial expansions and pseudodistributions.

Contribution

It introduces a novel method using orthogonal polynomial expansions and pseudodistributions to generate and analyze superoscillating and supergrowing bandlimited functions.

Findings

Explicit construction using Legendre polynomials in Fourier space

Ability to mimic arbitrary behaviors in finite intervals

Derived bounds on energy content in superoscillating regions

Abstract

We give a general strategy to construct superoscillating/growing functions using an orthogonal polynomial expansion of a bandlimited function. The degree of superoscillation/growth is controlled by an anomalous expectation value of a pseudodistribution that exceeds the band limit. The function is specified via the rest of its cumulants of the pseudodistribution. We give an explicit construction using Legendre polynomials in the Fourier space, which leads to an expansion in terms of spherical Bessel functions in the real space. The other expansion coefficients may be chosen to optimize other desirable features, such as the range of super behavior. We provide a prescription to generate bandlimited functions that mimic an arbitrary behavior in a finite interval. As target behaviors, we give examples of a superoscillating function, a supergrowing function, and even a discontinuous step function. We also look at the energy content in a superoscillating/supergrowing region and provide a bound that depends on the minimum value of the logarithmic derivative in that interval. Our work offers a new approach to analyzing superoscillations/supergrowth and is relevant to the optical field spot generation endeavors for far-field superresolution imaging.