# Generalization of Bandlimited Functions and Applications to Quantum Probability Distributions

## TL;DR

This paper extends the concept of bandlimited functions beyond Fourier transforms and applies it to quantum probability distributions, deriving bounds on wave functions and their properties.

## Contribution

The paper introduces a generalization of bandlimited functions to arbitrary representations and derives new inequalities and bounds for quantum wave functions and probability distributions.

## Key findings

- Explicit bounds on position wave functions and their derivatives are derived for bounded momentum wave functions.
- Bounds on momentum wave functions and probability distributions are obtained when position wave functions are bounded.
- Probability distribution bounds are derived for wave functions composed of a finite number of energy eigenfunctions.

## Abstract

Bandlimited functions are functions whose Fourier transform is confined to a finite band of frequencies. We generalize this concept to representations other than the Fourier transform and show that this leads to a variety of inequalities in arbitrary representations. Several special cases are considered, including frequency, dilation, and the chirplet transform, among others. Examples are given to illustrate each result. We apply the results to quantum mechanical wave functions and probability distributions. For bounded momentum wave functions, we obtain explicit bounds on the position wave function and its derivatives, as well as bounds on the position probability distribution. We also consider the dual problem in which the position wave function is bounded, as in the case of a particle in a box with an arbitrary wave function, and obtain bounds on the corresponding momentum wave function and momentum probability distribution. The case of wave functions that are sums of a finite number of energy eigenfunctions is also developed, and bounds on the associated probability distributions are obtained. A number of specific examples are considered, including a truncated Gaussian wave function and a quantum bump wave function.