Multi-exponential analysis might sound remote, but it touches our lives in many surprising ways, even if most people are unaware of just how important it is. For example, a substantial amount of effort in the field of signal processing is essentially dedicated to the analysis of multi-exponential functions whose exponents are complex. The analysis of exponential functions whose exponents are very near each other is directly linked to superresolution imaging. As for multi-exponential functions with real exponents, they are used to portray relaxation, chemical reactions, radioactivity, heat transfer, fluid dynamics, to name just a few.
Despite their computational efficiency and wide applicability, Fourier-based methods suffer some well-known limitations, such as a limited resolution and leakage in the frequency domain. They are also not well suited for aperiodic signals. Multi-exponential analysis on the other hand, may suffer from ill-posedness or ill-conditioning. The underlying method to reconstruct the exponential sum is already much older and goes back to de Prony. Both Fourier- and Prony-based methods make use of the Shannon-Nyquist theorem which states that the minimum sampling rate required to recover a signal is at least twice the maximum frequency present in the band-limited signal.
Recent achievements, allowing to circumvent the Shannon-Nyquist sampling rate, to regularize the multi-exponential analysis and to break the curse of dimensionality in case of higher-dimensional analyses, have unlocked new potentials.