Symmetry, Scaling, and Optimal Time-Frequency Concentration: Minimising the Heisenberg Uncertainty in Piecewise-Polynomial and Wavelet Dictionaries

Abstract

In this work, we introduce a hierarchy of function classes defined on a fixed compact interval, along with tailored uncertainty operators. We establish key properties of the associated uncertainty product, showing that it is invariant under scale and translation transformations. Notably, we prove that the infimum of the uncertainty within the asymmetric class is attained in the even subclass. Within two specific wavelet dictionaries, we identify the tent function as the unique minimiser of the time-frequency uncertainty, achieving a value of $U = \frac{3}{10}$. Additionally, we analyse the family of $p$-fold self-convolutions of the rectangle function, $\operatorname{rect}^{\{p\}}$, demonstrating that the uncertainty decreases monotonically towards the Heisenberg bound $ \frac{1}{4} $ as $p \to \infty$. These findings unify and explain various empirical observations from the literature on adaptive wavelet design and Gabor frame stability, and suggest a principled approach to constructing dictionaries with provably optimal joint localisation properties.