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Orbits of bounded bijective operators and Gabor frames

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Abstract

This paper is a contribution to frame theory. Frames in a Hilbert space are generalizations of orthonormal bases. In particular, Gabor frames of \(L^2(\mathbb {R})\), which are made of translations and modulations of one or more windows, are often used in applications. More precisely, the paper deals with a question posed in the last years by Christensen and Hasannasab about the existence of overcomplete Gabor frames, with some ordering over \(\mathbb {Z}\), which are orbits of bounded operators on \(L^2(\mathbb {R})\). Two classes of overcomplete Gabor frames which cannot be ordered over \(\mathbb {Z}\) and represented by orbits of operators in \(GL(L^2(\mathbb {R}))\) are given. Some results about operator representation are stated in a general context for arbitrary frames, covering also certain wavelet frames.

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References

  1. Aguilera, A., Cabrelli, C., Carbajal, D., Paternostro, V.: Dynamical Sampling for Shift-preserving Operators. arXiv:1912.10348 (2019)

  2. Aldroubi, A., Cabrelli, C., Molter, U., Petrosyan, A.: Local-to-global frames and applications to dynamical sampling problem. arXiv:1909.02987 (2019)

  3. Aldroubi, A., Cabrelli, C., Cakmak, A.F., Molter, U., Petrosyan, A.: Iterative actions of normal operators. J. Funct. Anal. 272, 1121–1146 (2017)

    Article  MathSciNet  Google Scholar 

  4. Aldroubi, A., Cabrelli, C., Molter, U., Tang, S.: Dynamical Sampling. Appl. Comput. Harmon. Anal. 42, 378–401 (2017)

    Article  MathSciNet  Google Scholar 

  5. Aldroubi, A., Petrosyan, A.: Dynamical sampling and systems from iterative actions of operators. In: Pesenson, I., Le Gia, Q., Mayeli, A., Mhaskar, H., Zhou, D.X. (eds.) Frames and other bases in abstract and function spaces Applied and Numerical Harmonic Analysis. Birkhäuser, Boston (2017)

    Google Scholar 

  6. Balan, R., Casazza, P., Heil, C., Landau, Z.: Deficits and excesses of frames. Adv. Comput. Math. 18, 93–116 (2002)

    Article  MathSciNet  Google Scholar 

  7. Cabrelli, C., Molter, U., Paternostro, V., Philipp, F.: Dynamical Sampling on finite index sets, arXiv:1702.03384

  8. Cabrelli, C., Molter, U., Paternostro, V., Philipp, F.: Finite sensor dynamical sampling. In: Proceedings of the 12th International Conference on Sampling Theory and Applications (SampTA), pp. 50–54 (2017)

  9. Cabrelli, C., Molter, U., Suarez, D.: Multi-orbital frames through model spaces, arXiv:1908.11011 (2019)

  10. Christensen, O.: An Introduction to Frames and Riesz Bases. Birkhäuser, Boston (2016)

    MATH  Google Scholar 

  11. Christensen, O., Hasannasab, M.: Approximate frame representations via iterated operator systems. Stud, Math (2019). in press)

    MATH  Google Scholar 

  12. Christensen, O., Hasannasab, M.: Frame representations via suborbits of bounded operators. In: Proceedings of the 13th International Conference on Sampling Theory and Applications (SampTA) (2019)

  13. Christensen, O., Hasannasab, M.: Frame properties of systems arising via iterative actions of operators. Appl. Comput. Harmon. Anal. 46(3), 664–673 (2019)

    Article  MathSciNet  Google Scholar 

  14. Christensen, O., Hasannasab, M.: Operator representations of frames. In: Proceedings of the 12th International Conference on Sampling Theory and Applications (SampTA), pp. 207–211 (2017)

  15. Christensen, O., Hasannasab, M.: Operator representations of frames: boundedness, duality, and stability. Integr. Equ. Oper. Theory 88, 483–499 (2017)

    Article  MathSciNet  Google Scholar 

  16. Christensen, O., Hasannasab, M.: Frames, operator representations, and open problems. In: Böttcher, A., Potts, D., Stollmann, P., Wenzel, D. (eds.) The Diversity and Beauty of Applied Operator Theory. Operator Theory: Advances and Applications, vol 268. Birkhäuser, Cham

  17. Christensen, O., Hasannasab, M., Philipp, F.: Frame properties of operator orbits. Math, Nach (2019). in press)

    MATH  Google Scholar 

  18. Christensen, O., Hasannasab, M., Rashidi, E.: Dynamical sampling and frame representations with bounded operators. J. Math. Anal. Appl. 463(2), 634–644 (2018)

    Article  MathSciNet  Google Scholar 

  19. Feichtinger, H.G., Strohmer, T. (eds.): Gabor Analysis And Algorithms: Theory and Applications. Birkhäuser Boston, Boston (1998)

    MATH  Google Scholar 

  20. Gröchenig, K.: Foundations of Time-Frequency Analysis. Birkhäuser, Boston (2001)

    Book  Google Scholar 

  21. Heil, C.: A Basis Theory Primer, Expanded edn. Birkhäuser, Boston (2011)

    Book  Google Scholar 

  22. Philipp, F.: Bessel orbits of normal operators. J. Math. Anal. Appl. 448, 767–785 (2017)

    Article  MathSciNet  Google Scholar 

  23. Reed, M., Simon, B.: Methods of Modern Mathematical Physics II. Self-adjointness. Academic Press, New York, Fourier Analysis (1972)

    MATH  Google Scholar 

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Acknowledgements

This work was partially supported by the “Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni” (GNAMPA - INdAM).

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  1. Dipartimento di Matematica e Informatica, Università degli Studi di Palermo, Via Archirafi 34, 90123, Palermo, Italy

    Rosario Corso

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  1. Rosario Corso
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Correspondence to Rosario Corso.

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Corso, R. Orbits of bounded bijective operators and Gabor frames. Annali di Matematica 200, 137–148 (2021). https://doi.org/10.1007/s10231-020-00988-1

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  • DOI: https://doi.org/10.1007/s10231-020-00988-1

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