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Uncertainty-type relationships for periodic functions

Bastiaans, M.J., Alonso Arevalo, M.A., Philips, W.

Published Jan 1, 2004• 0 citations
OpenAlex

Abstract

Based on five different mappings of a function on a circle (i.e. a periodic function) to a function on a line or a line segment (i.e. a non-periodic function), measures for the width of a periodic function (expressed in terms of centered second-order moments) are defined, and the associated uncertainty-type relationships are derived, using a generalized version of the Cauchy-Schwarz inequality.

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Based on five different mappings of a function on a circle (i.e. a periodic function) to a function on a line or a line segment (i.e. a non-periodic function), measures for the width of a periodic function (expressed in terms of centered second-order moments) are defined, and the associated uncertainty-type relationships are derived, using a generalized version of the Cauchy-Schwarz inequality.
A physicist’s intuition in Fourier theory is generally established from the parallels between Fourier series and transforms. Remarkably, one element of this theory that is especially significant in physics, namely the uncertainty principle, is never treated for Fourier series. We resolve this by first showing that a natural measure of spread for a periodic distribution follows simply upon regarding the distribution as a mass density on a ring. Even though the centroid of this ring is expressed in terms of just a first moment, its distance from the geometric center gives a close analog of variance. We then derive direct analogs of the uncertainty principle for both the Fourier series of a continuous periodic function as well as the fast Fourier transform of discrete data. The results have similar applications to those of the standard uncertainty principle.
The conventional Fourier transform has a well-known uncertainty relation that is defined in terms of the first and second moments of both a function and its Fourier transform. It is also well known that Gaussian functions, when translated to an arbitrary centre and supplemented by a linear phase factor, provide a complete set of minimum uncertainty states (MUSs) that exactly achieves the lower bound set by this uncertainty relation. A similarly general set of MUSs and uncertainty relations are derived here for discrete and/or periodic generalizations of the Fourier transform, namely for the discrete Fourier transform and the Fourier series. These extensions require a modified definition for the width of a periodic distribution, and they lead to more complex uncertainty relations that turn out to depend on the centroid location and mean frequency of the distribution. The derivations lead to novel generalizations of Hermite–Gaussian functions and, like Gaussians, the MUSs can play a special role in a range of Fourier applications.