Refining Heisenberg's principle: A greedy approximation of step functions with triangular waveform dictionaries

In this paper, we consider a step function characterized by a real-valued sequence and its linear expansion representation constructed via the matching pursuit (MP) algorithm. We utilize a waveform dictionary based on the triangular function as part of this algorithm and representation. The waveform dictionary is comprised of waveforms localized in the time–frequency domain. In view of this, we prove that the triangular waveforms are more efficient than the rectangular waveforms used in a prior study by achieving a product of variances in the time–frequency domain closer to the lower bound of the Heisenberg Uncertainty Principle. We provide a MP algorithm solvable in polynomial time, contrasting the common exponential time when using Gaussian windows. We apply this algorithm on simulated data and real GDP data from 1947–2024 to demonstrate its application and efficiency.