Full Spark Frames in Chebotarev's Theorem and Their Application in Signal Reconstruction

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.46698/e5734-0206-5941-z Full Spark Frames in Chebotarev's Theorem and Their Application in Signal Reconstruction Novikov, S. Ya. Vladikavkaz Mathematical Journal 2026. Vol. 28. Issue 2. Abstract: Full Spark Frames in finite-dimensional Euclidean (Unitary) spaces are complete systems in which each subsystem with a number of vectors equal to the dimension of the space is complete. Chebotarev's theorem on the discrete Fourier transform matrix turned out to be historically the first theorem on equi-normed full spark Parseval frames. When the number of measurement vectors is minimal, full spark frames ensure the injectivity of a nonlinear operator that converts the signal into measurement modules. This article provides an overview of the applications of frames in the theory of neural networks. In particular, it establishes criteria for the injectivity of ReLU-layers. The article also explores the existence of full spark frames in infinite-dimensional Hilbert spaces. In an infinite-dimensional Hilbert space, full spark systems are those in which every infinite subsystem is complete. If the norms of the frame elements are separated from zero, then such a frame cannot have a full spark. It is unknown whether there are general frames with a full spark. The existence of equal-norm Parseval frames in infinite-dimensional Hilbert spaces is considered. Keywords: frame, spark, injective operator, measurement vectors, measurement modules, ReLU-layers Funding: The work was carried out within the framework of the development program of the Scientific and Educational Mathematical Center of the Volga Federal District, agreement No. 075-02-2025-1791. Language: Russian Download the full text Download JATS XML For citation: Novikov, S. Ya. Full Spark Frames in Chebotarev's Theorem and Their Application in Signal Reconstruction, Vladikavkaz Math. J., 2026, vol. 28, no. 2, pp. 108-117 (in Russian). DOI 10.46698/e5734-0206-5941-z + References 1. Novikov, S. Ya. and Sevost'yanova, V. V. Maltsev Equal-Norm Tight Frames, Izvestiya: Mathematics, 2022, vol. 86, no. 4, pp. 770-781. DOI: 10.4213/im9137e. 2. Alexeev, B., Cahill, J. and Mixon, D. G. Full Spark Frames, Journal of Fourier Analysis and Applications, 2012, vol. 18, no. 6, pp. 1167-1194. DOI: 10.1007/s00041-012-9235-4. 3. Chebotarev, N. G. Mathematical Autobiography, Russian Mathematical Surveys, 1948, vol. 3, no. 4, pp. 3-66 (in Russian). 4. Balan, R., Casazza, P. and Edidin, D. On Signal Reconstruction Without Phase, Applied and Computational Harmonic Analysis, 2006, vol. 20, no. 3, pp. 345-356. DOI: 10.1016/J.ACHA.2005.07.001. 5. Izbiakov, I. M., Novikov, S. Ya. and Terekhin, P. 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An Introduction to Frames and Riesz Bases, New York, Birkhauser/Springer, 2016, 704 p. 11. Marcus, A. W., Spielman, D. A. and Srivastava, N. Interlacing Families II: Mixed Characteristic Polynomials and the Kadison-Singer Problem, Annals of Mathematics, 2015, vol. 182, no. 1, pp. 327-350. DOI: 10.4007/annals.2015.182.1.8. 12. Aldroubi, A., Cabrelli, C., Molter, U. and Tang, S. Dynamical Sampling, Applied and Computational Harmonic Analysis, 2017, vol. 42, no. 3, pp. 378-401. DOI: 10.1016/j.acha.2015.08.014. 13. Christensen, O., Hasannasab, M., Philipp, F., and Stoeva, D. The Mystery of Carleson Frames, Applied and Computational Harmonic Analysis, 2024, vol. 72, Article ID 101659, 5 p. DOI: 10.1016/j.acha.2024.101659. 14. Krishtal, I. and Miller, B. Letter to the Editor. Demystifying Carleson Frames, Applied and Computational Harmonic Analysis, 2026, vol. 80, Article ID 101811, 8 p. The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. 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