Mathematicians Dr. Arvin Lamando of the University of the Philippines – Diliman College of Science’s Institute of Mathematics (UPD-CS IM) and Dr. Henry McNulty of the Norwegian University of Science and Technology have found a new way to understand mathematical “machines” called operators, which are key to quantum mechanics and signal processing. Their study shows that even the most intricate of these operators can be broken down into simpler parts and then reconstructed, offering new insights into quantum systems and technologies.

“My research is mostly within the area of mathematics called ‘harmonic analysis,’” Dr. Lamando said. “Can we always decompose arbitrary signals f as a sum of pure frequencies (sines and cosines)? The Fourier transform answers this.”

We can think of the signal as a musical chord. The Fourier transform breaks the sound down into its individual pure notes. And just like how we can replay the chord by pressing those same notes together on a piano, we can also reconstruct “abstract signals” from its “pure frequencies.”

“While harmonic analysis has historical roots in real-world applications, the ideas related to the Fourier transform turned out to be very amenable to abstraction; and it is surprising that it has connections to different branches of abstract mathematics,” he explained.

As classical harmonic analysis deals with signals and their frequencies, quantum harmonic analysis applies similar ideas to operators. This field studies operators that follow specific mathematical rules used when translating classical physics into quantum physics. 

“We also introduced another notion, called the ‘modulation’ of an operator in the phase space. This notion is consistent with the main themes of quantum harmonic analysis: in fact, the operator Fourier transform of operator modulation results in a translated operator Fourier transform,” he said. In their study, the mathematicians focused on operators that remain unchanged, or invariant, even when translated or modulated over lattices on the phase space. 

“We have shown that these operators possess properties analogous to the classical case,” Dr. Lamando shared, adding that he and Dr. McNulty used a mathematical framework called the Heisenberg module to better understand and describe these operators.

The mathematicians found that these invariant operators can be closely approximated using much simpler operators called finite-rank operators, which can roughly be interpreted to mean that their outputs can be described using only a finite number of dimensions. Their results bridge abstract algebraic ideas to concrete structures in quantum mathematics.

Their research, “On Modulation and Translation Invariant Operators and the Heisenberg Module,” was included in the Journal of Fourier Analysis and Applications, which publishes articles with topics that range from abstract harmonic analysis and group representation theory to real world applications and partial differential equations.

References:

Lamando, A., & McNulty, H. (2025). On modulation and translation invariant operators and the heisenberg module. Journal of Fourier Analysis and Applications, 31(4). https://doi.org/10.1007/s00041-025-10176-5