Superoscillations with arbitrary polynomial shape

TL;DR

This paper introduces a method to construct superoscillatory functions that can approximate any polynomial shape within a fixed interval with high accuracy, using bandlimited envelope functions with specific smoothness properties.

Contribution

It provides a novel approach to generate superoscillations with arbitrary polynomial shapes by combining polynomials with specially designed bandlimited envelope functions.

Findings

Superoscillatory functions can approximate arbitrary polynomials with small error.

The method works for polynomials of arbitrarily high order.

Smooth envelope functions enable high-order polynomial approximation.

Abstract

We present a method for constructing superoscillatory functions the superoscillatory part of which approximates a given polynomial with arbitrarily small error in a fixed interval. These functions are obtained as the product of the polynomial with a sufficiently flat, bandlimited envelope function whose Fourier transform has at least N-1 continuous derivatives and an N-th derivative of bounded variation, N being the order of the polynomial. Polynomials of arbitrarily high order can be approximated if the Fourier transform of the envelope is smooth, i.e. a bump function.