In today’s data-driven world, where vast volumes of information are generated across scientific, medical, and technological domains, the challenge of extracting meaningful insights from limited, noisy, or high-dimensional measurements has become more pressing than ever. A central problem in this context is the identification of sparse, low-complexity model representations – specifically, models that can accurately describe complex phenomena using the fewest possible parameters or measurements. This is particularly critical in the analysis of multi-exponential signals, where the goal is to recover a small number of exponential components (with real or complex exponents) from a minimal set of observations – often referred to as “probes” or “samples”. The acquisition of such measurements is frequently constrained by practical limitations. In medical imaging, for instance, scanning time directly impacts patient comfort and throughput. In remote sensing or radio astronomy, data collection involves expensive instrumentation and limited bandwidth. In industrial testing, measurement costs can be prohibitive. As a result, researchers are increasingly forced to work with datasets that are not only limited in size but also corrupted by noise, making accurate reconstruction a non-trivial task. This has elevated the need for robust, efficient, and theoretically grounded methods for exponential analysis – methods that can deliver high-fidelity results even under severe data scarcity and noise contamination. Multi-exponential analysis, though seemingly abstract, plays a surprisingly central role in numerous everyday technologies and scientific disciplines. At its core, it involves decomposing a signal into a sum of exponentials – functions of the form

f(t) = ∑_k=1^r c_k,e^λ_kt,

where c_k and λ_k may be real or complex.

When the exponents are complex, such models are essential in analyzing oscillatory signals - common in digital signal processing, time series forecasting, and spectral estimation. These techniques are used to extract frequencies, damping rates, and amplitudes from noisy data, forming the backbone of applications ranging from audio and speech processing to financial market modeling and biomedical signal analysis.

When the exponents are real, multi-exponential models describe fundamental physical processes: relaxation dynamics in materials science, decay rates in radioactive isotopes, reaction kinetics in chemistry, heat diffusion in engineering systems, and fluid flow in environmental modeling. These models are not just theoretical constructs - they are indispensable tools for understanding and predicting real-world behavior across physics, biology, and engineering.

The mathematical and computational challenges of multi-exponential analysis are deeply intertwined with several advanced areas of computational science. The problem naturally connects to structured matrix theory – where the Hankel or Vandermonde structure of the data matrix encodes the exponential form – enabling efficient algorithms via rank minimization and low-rank approximation. Rational approximation theory provides powerful tools for modeling the Z-transformed signal data as a ratio of polynomials, thereby linking the λ_k and c_k to poles and residues. Sparse interpolation techniques allow for the recovery of parameters from few samples, while scale-and-shift invariance principles offer robustness to signal transformations. Tensor decomposition methods and multivariate quadrature extend these ideas to multi-dimensional data, where the curse of dimensionality poses a fundamental challenge, and advanced techniques including sparsity models are of high importance. Furthermore, non-convex optimization plays a crucial role, as the parameter estimation problem often leads to non-convex cost functions with multiple local minima – requiring sophisticated initialization and convergence strategies. Subdivision methods further extend the reach of these techniques into geometric and directional data analysis.

The impact of multi-exponential analysis extends far beyond theory. It is foundational in a wide array of engineering and industrial applications: direction-of-arrival (DOA) estimation in radar and wireless communications, high-resolution remote sensing and satellite imaging, antenna array design for 5G and beyond, digital image reconstruction and superresolution, precision metrology in manufacturing, radio astronomy for detecting faint cosmic signals, and magnetic resonance imaging (MRI), where fast, accurate reconstruction enables shorter scan times and improved diagnostics. These technologies are not only advancing scientific discovery but are also addressing major societal challenges – improving healthcare outcomes, enabling sustainable energy systems, enhancing transportation safety, supporting space exploration, and strengthening global communication networks.

Given this broad relevance, this Dagstuhl Seminar “From Sparse Interpolation to Signal Processing: New Synergies” (25281) aimed to serve as a vital interdisciplinary forum, bringing together experts from computational harmonic analysis, numerical linear algebra, computer algebra, nonlinear approximation theory, digital signal processing, as well as partners from industry. By fostering dialogue between researchers who have developed similar concepts in isolation, we hope to catalyze cross-fertilization, unify methodologies, and identify shared challenges and opportunities. The goal has been not only to advance the theoretical foundations of exponential analysis but also to accelerate the development of next-generation algorithms that are faster, more robust, and scalable – ultimately enabling breakthroughs in data science, engineering, and beyond.

The talks of this Seminar have been organized with emphasis to the following 6 main topics:

• Generalisations of exponential analysis
  (G. Plonka-Hoch, Y. Segman, H. Mhaskar, R. O’Dowd, D. Potts, and J. Prestin),

• Exponential analysis and structured matrices
  (A. Matos, H. Liang, M. Ishteva, T. Sauer, and A. Iske)

• Exponential analysis in computational science
  (W.-S. Lee, J. Gielis, D. Li, A. Beutler, and R. Beinert)

• Exponential analysis in quadrature and subdivision
  (A. Cuyt, T. Perez, M. Cotronei, and M. Piñar)

• Exponential analysis in engineering
  (D. de Villiers, D. Davidson, J. Gilmore, N. Diab, and A. Terui)

• Exponential analysis and computer algebra
  (J. Gerhard, E. Kaltofen, B. Grenet, and P. Giorgi)

The seminar began on Monday with overview lectures on the first five main topics of the meeting, enabling all participants to quickly gain an entry into the fascinating open interdisciplinary challenges surrounding exponential analysis.

The talk topics on Tuesday covered connections between exponential analysis and sparse approximation, structured matrices, as well as problems in signal analysis and signal separation.

On Wednesday, the topics focused on multivariate integration and subdivision. The free afternoon was used for a short trip to Bernkastel-Kues by several participants and provided the opportunity for physical exercise and lively discussions on the seminar topics.

Thursday was dedicated to various application-oriented topics in exponential analysis. The discussions ranged from sparse models in biology and confocal microscopy to questions concerning the efficient measurement of mutual coupling terms in linear arrays.

Finally, the close connections to special problems in computer algebra, as for example, a quasi-linear time sparse interpolation algorithm over the integers or the fast interpolation and multiplication of unbalanced polynomials, have been the main topic on the final day.

We would like to highlight that this seminar builds upon the 2015 Dagstuhl Seminar 15251, titled “Sparse Modelling and Multi-Exponential Analysis” and the 2022 Dagstuhl Seminar 22221 “Exponential Analysis: Theoretical Progress and Technological Innovation”.

The discussions held during the 2015 event sparked numerous fruitful collaborations, including the successful Horizon 2020 RISE project EXPOWER – short for “Exponential Analysis Empowering Innovation” (Grant Agreement No. 101008231, running from 2021-2026), with Annie Cuyt as the coordinator. This project exemplifies how foundational research in exponential analysis can translate into impactful, cross-sectoral innovation.

In October 2025, three months after this meeting, we submitted a new proposal to the Call: Horizon-MSCA-2025-SE-01 (MSCA Staff Exchanges 20225) with the topic TRESUR – “Building synergies between industry and mathematical topics in sparse approximation and recovery”, coordinated by Tereza Pérez, in close collaboration with A. Cuyt, W.-S. Lee, A. Matos, M. Piñar, D. de Villiers, G. Plonka-Hoch and several further participants of this Dagstuhl Seminar.

Our experience confirms that Dagstuhl Seminars serve as timely and transformative forums for scientific exchange. They create fertile ground for new partnerships, stimulate interdisciplinary thinking, and unlock novel research directions. In light of rapid advances in both theoretical methods and real-world applications, there is a growing need to strengthen the bridge between cutting-edge mathematical developments and practical industrial challenges. This seminar, and our ongoing efforts, aim to foster such connections – ensuring that theoretical progress continues to inspire and inform real-world innovation.

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Annie Cuyt, Dirk de Villiers, Wen-shin Lee, Ana C. Matos, and Gerlind Plonka-Hoch