On the Relationship between Uniqueness Sets and Interpolation Sets in Functional Quasinormed Spaces
Résumé
Given a functional quasinormed space we obtain conditions for a uniformly discrete set to to be either a uniqueness set or an interpolation set. We apply these results to Paley-Wiener spaces. We also obtain new results on both the refinement of stable sampling sets and the extension of stable interpolation sets in quasinormed spaces.
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Références
[1] Avantaggiati, A., Loreti, P., Velluci, P., An Explicit Bound for Stability of Sinc Bases, Proceedings of the 12th International Conference on Informatics in Control, Automation and Robotics 1 (2015), 473–480, doi 10.5220/0005512704730480.
[2] Avantaggiati, A., Loreti, P., Velluci, P., Kadec-1/4 Theorem for Sinc Bases, (2016) ArXiv 1603.08762v1.
[3] Avdonin, S. A., On the question of Riesz bases of exponential functions in L2, Vestnik Leningrad Univ. 13 (1974), 5–12 (Russian). English translation in Vestnik Leningrad Univ. Math. 7 (1979), 203–211.
[4] Beurling, A., Balayage of Fourier-Stieltjes Transforms. In: The collected Works of Arne Beurling, vol. 2, Harmonic Analysis., Birkh¨auser, Boston, 1989.
[5] Beurling, A., Interpolation for an interval in R. In The collected Works of Arne Beurling 2, Harmonic Analysis, Birkhuser, Boston (1989), 351–365.
[6] Beurling, A., Malliavin, P. On the closure of characters and the zeros of entire functions., Acta Math., 118 (1967), 79–93.
[7] Kadec, M. I., The exact value of the Paley-Wiener constant, Sov. Math. Dokl. 5 (1964), 559–561.
[8] Levinson, N., Gap and Density Theorems, Colloquium Publications 26, First Edition, American Mathematical Soc., 1940.
[9] López Nicolás, J. A., Irregular Stable Sampling and Interpolation in Functional Normed Spaces, Bol. Soc. Paran. Mat. 40 (2019), 1–13, doi 10.5269/bspm.45497.
[10] López Nicolás, J. A., On Stable Sampling and Interpolation in Bernstein Spaces, Revista Colombiana de Matemáticas 56 (2) (2022), 1–13, doi 10.15446/recolma.v56n2.108383.
[11] López Nicolás, J. A., Recombination of Stable Sampling Sets and Stable Interpolation Sets in Functional Quasinormed Spaces, Bol. Soc. Paran. Mat. 42 (2024), 1–14, doi 10.5269/bspm.62925.
[12] Mitkovski, M., Wick, B. D., On the Uniqueness Sets in the Fock Space arXiv-org. preprint, arXiv:1306.0318 [math.CV], 2013, doi 10.48550/ARXIV.1306.0318.
[13] Neumann, J. von, The Mathematical Foundations of Quantum Mechanics Princeton University Press, NJ, 1955.
[14] Olevskii, A., Ulanovskii, A., Universal Sampling and Interpolation of Band-limited Signals Geometric and Functional Analysis 18 (3) (2008), 1029–1052, doi 10.1007/s00039-008-0674-7.
[15] Olevskii, A., Ulanovskii, A., Uniqueness sets for unbounded spectra Comptes Rendus Mathematique 349 (11) (2011), 679–681, doi 10.1016/j.crma.2011.05.010.
[16] Olevskii, A., Ulanovskii, A., Discrete Uniqueness Sets for Functions with Spectral Gaps Russian Academy of Sciences Sbornik Mathematics 208 (6) (2016), 43–53, doi 10.1070/SM8837.
[17] Olevskii, A., Ulanovskii, A., Functions with Disconnected Spectrum, University Lecture Series, First Edition, American Mathematical Society, 2016.
[18] Olevskii, A., Ulanovskii, A., On the duality between sampling and interpolation Analysis Mathematica 42 (2016), 43–53, doi 10.1007/s10476-016-0104-2.
[19] Paley, R., Wiener, N. Fourier transforms in the complex domain, Amer. Math. Soc. Colloquium Publications 19 (1934).
[20] Plancherel, M., Pólya, G., Fonctions entiéres et intégrales de Fourier multiples (seconde partie) Comment. Math. Helv. 10 (1) (1937), 110–163, doi 10.1007/bf01214286.
[21] Sedleckii, A. M., Nonharmonic Fourier Series, Sib. Math. J. 12 (1971a), 793–802.
[22] Sedletskii, A. M., Nonharmonic Analysis, J. Math. Sc. 116 (5) (2009), 3551–3619.
[23] Ullrich, D., Divided differences and systems of nonharmonic Fourier series, Proc. Amer. Math. Soc. 80 (1980), 47–57.
[24] Young, R. M., An introduction to Nonharmonic Fourier Series, Academic Press, Second Edition, 2001.
[2] Avantaggiati, A., Loreti, P., Velluci, P., Kadec-1/4 Theorem for Sinc Bases, (2016) ArXiv 1603.08762v1.
[3] Avdonin, S. A., On the question of Riesz bases of exponential functions in L2, Vestnik Leningrad Univ. 13 (1974), 5–12 (Russian). English translation in Vestnik Leningrad Univ. Math. 7 (1979), 203–211.
[4] Beurling, A., Balayage of Fourier-Stieltjes Transforms. In: The collected Works of Arne Beurling, vol. 2, Harmonic Analysis., Birkh¨auser, Boston, 1989.
[5] Beurling, A., Interpolation for an interval in R. In The collected Works of Arne Beurling 2, Harmonic Analysis, Birkhuser, Boston (1989), 351–365.
[6] Beurling, A., Malliavin, P. On the closure of characters and the zeros of entire functions., Acta Math., 118 (1967), 79–93.
[7] Kadec, M. I., The exact value of the Paley-Wiener constant, Sov. Math. Dokl. 5 (1964), 559–561.
[8] Levinson, N., Gap and Density Theorems, Colloquium Publications 26, First Edition, American Mathematical Soc., 1940.
[9] López Nicolás, J. A., Irregular Stable Sampling and Interpolation in Functional Normed Spaces, Bol. Soc. Paran. Mat. 40 (2019), 1–13, doi 10.5269/bspm.45497.
[10] López Nicolás, J. A., On Stable Sampling and Interpolation in Bernstein Spaces, Revista Colombiana de Matemáticas 56 (2) (2022), 1–13, doi 10.15446/recolma.v56n2.108383.
[11] López Nicolás, J. A., Recombination of Stable Sampling Sets and Stable Interpolation Sets in Functional Quasinormed Spaces, Bol. Soc. Paran. Mat. 42 (2024), 1–14, doi 10.5269/bspm.62925.
[12] Mitkovski, M., Wick, B. D., On the Uniqueness Sets in the Fock Space arXiv-org. preprint, arXiv:1306.0318 [math.CV], 2013, doi 10.48550/ARXIV.1306.0318.
[13] Neumann, J. von, The Mathematical Foundations of Quantum Mechanics Princeton University Press, NJ, 1955.
[14] Olevskii, A., Ulanovskii, A., Universal Sampling and Interpolation of Band-limited Signals Geometric and Functional Analysis 18 (3) (2008), 1029–1052, doi 10.1007/s00039-008-0674-7.
[15] Olevskii, A., Ulanovskii, A., Uniqueness sets for unbounded spectra Comptes Rendus Mathematique 349 (11) (2011), 679–681, doi 10.1016/j.crma.2011.05.010.
[16] Olevskii, A., Ulanovskii, A., Discrete Uniqueness Sets for Functions with Spectral Gaps Russian Academy of Sciences Sbornik Mathematics 208 (6) (2016), 43–53, doi 10.1070/SM8837.
[17] Olevskii, A., Ulanovskii, A., Functions with Disconnected Spectrum, University Lecture Series, First Edition, American Mathematical Society, 2016.
[18] Olevskii, A., Ulanovskii, A., On the duality between sampling and interpolation Analysis Mathematica 42 (2016), 43–53, doi 10.1007/s10476-016-0104-2.
[19] Paley, R., Wiener, N. Fourier transforms in the complex domain, Amer. Math. Soc. Colloquium Publications 19 (1934).
[20] Plancherel, M., Pólya, G., Fonctions entiéres et intégrales de Fourier multiples (seconde partie) Comment. Math. Helv. 10 (1) (1937), 110–163, doi 10.1007/bf01214286.
[21] Sedleckii, A. M., Nonharmonic Fourier Series, Sib. Math. J. 12 (1971a), 793–802.
[22] Sedletskii, A. M., Nonharmonic Analysis, J. Math. Sc. 116 (5) (2009), 3551–3619.
[23] Ullrich, D., Divided differences and systems of nonharmonic Fourier series, Proc. Amer. Math. Soc. 80 (1980), 47–57.
[24] Young, R. M., An introduction to Nonharmonic Fourier Series, Academic Press, Second Edition, 2001.
Publiée
2025-08-24
Numéro
Rubrique
Research Articles
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