Continuous Boostlet Transform and Associated Uncertainty Principles

Abstract

The Continuous Boostlet Transform (CBT) is introduced as a powerful tool for analyzing spatiotemporal signals, particularly acoustic wavefields. Overcoming the limitations of classical wavelets, the CBT leverages the Poincaré group and isotropic dilations to capture sparse features of natural acoustic fields. This paper presents the mathematical framework of the CBT, including its definition, fundamental properties, and associated uncertainty principles, such as Heisenberg's, logarithmic, Pitt's, and Nazarov's inequalities. These results illuminate the trade-offs between time and frequency localization in the boostlet domain. Practical examples with constant and exponential functions highlight the CBT's adaptability. With applications in radar, communications, audio processing, and seismic analysis, the CBT offers flexible time-frequency resolution, making it ideal for non-stationary and transient signals, and a valuable tool for modern signal processing.

Figures (1)

• Figure 1: Interactive surface plot of the Boostlet transform magnitude $|B_{\varphi}f(c,\alpha,\tau)|$ in 2D Minkowski space, showing how the transform varies under different Lorentz boost parameters ($\alpha$), dilation factors ($c$), and spacetime translations ($\tau = (\tau_{s}, \tau_{t})$). The slider controls enable exploration of boost-induced hyperbolic deformation effects and their impact on the localization of boostlet energy in spacetime.