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Frame Expansions of Test Functions, Tempered Distributions, and Ultradistributions

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Abstract

The paper is devoted to frame expansions in Fréchet spaces. First we review some results which concern series expansions in general Fréchet spaces via Fréchet and General Fréchet frames. Then we present some new results on series expansions of tempered distributions and ultradistributions, and the corresponding test functions, via localized frames and coefficients in appropriate sequence spaces.

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  • DOI: 10.1007/978-3-030-04459-6_44
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References

  1. A. Aldroubi, Q. Sun, W. Tang. p-Frames and shift invariant subspaces of L p. J. Fourier Anal. Appl. 7(1), 1–21 (2001)

    Google Scholar 

  2. R. Balan, P.G. Casazza, C. Heil, Z. Landau, Density, overcompleteness, and localization of frames. I: Theory. J. Fourier Anal. Appl. 12(2), 105–143 (2006)

    MathSciNet  MATH  Google Scholar 

  3. R. Balan, P.G. Casazza, C. Heil, Z. Landau, Density, overcompleteness, and localization of frames. II: Gabor systems. J. Fourier Anal. Appl. 12(3), 307–344 (2006)

    MathSciNet  MATH  Google Scholar 

  4. N.K. Bari, Biorthogonal systems and bases in Hilbert space, in Mathematics. Vol. IV. Uch. Zap. Mosk. Gos. Univ., vol. 148 (Moscow University Press, Moscow, 1951), pp. 69–107

    Google Scholar 

  5. J. Bonet, C. Fernández, A. Galbis, J.M. Ribera, Shrinking and boundedly complete Schauder frames in Fréchet spaces. J. Math. Anal. Appl. 410(2), 953–966 (2014)

    MathSciNet  MATH  CrossRef  Google Scholar 

  6. J. Bonet, C. Fernández, A. Galbis, J.M. Ribera, Frames and representing systems in Fréchet spaces and their duals. Banach J. Math. Anal. 11(1), 1–20 (2017)

    MathSciNet  MATH  CrossRef  Google Scholar 

  7. P.G. Casazza, D. Han, D.R. Larson, Frames for Banach spaces. Contemp. Math. 247, 149–182 (1999)

    MathSciNet  MATH  CrossRef  Google Scholar 

  8. P. Casazza, O. Christensen, D.T. Stoeva, Frame expansions in separable Banach spaces. J. Math. Anal. Appl. 307, 710–723 (2005)

    MathSciNet  MATH  CrossRef  Google Scholar 

  9. Y. Chen, M. Signahl, J. Toft, Factorizations and singular value estimates of operators with Gelfand-Shilov and Pilipović kernels. J. Fourier Anal. Appl. 24(3), 666–698 (2018)

    MathSciNet  MATH  CrossRef  Google Scholar 

  10. O. Christensen, An Introduction to Frames and Riesz Bases. Applied and Numerical Harmonic Analysis, second expanded edition (Birkhäuser, Boston, 2016)

    Google Scholar 

  11. R.J. Duffin, A.C. Schaeffer, A class of nonharmonic Fourier series. Trans. Am. Math. Soc. 72, 341–366 (1952)

    MathSciNet  MATH  CrossRef  Google Scholar 

  12. A.J. Duran, Laguerre expansions of tempered distributions and generalized functions. J. Math. Anal. Appl. 150(1), 166–180 (1990)

    MathSciNet  MATH  CrossRef  Google Scholar 

  13. A.J. Duran, Laguerre expansions of Gel’fand-Shilov spaces. J. Approx. Theory 74, 280–300 (1993)

    MathSciNet  MATH  CrossRef  Google Scholar 

  14. C. Fernandez, A. Galbis, J. Toft, The Bargmann transform and powers of harmonic oscillator on Gelfand-Shilov subspaces. Revista de la Real Academia de Ciencias Exactas, Fasicas y Naturales. Serie A. Matematicas 111(1), 1–13 (2017)

    Google Scholar 

  15. H.G. Feichtinger, K. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions. I. J. Funct. Anal. 86(2), 307–340 (1989)

    MathSciNet  MATH  CrossRef  Google Scholar 

  16. H.G. Feichtinger, K. Gröchenig, Banach spaces related to integrable group representations and their atomic decompositions II. Monatsh. Math. 108(2–3), 129–148 (1989)

    MathSciNet  MATH  CrossRef  Google Scholar 

  17. M. Fornasier, K. Gröchenig, Intrinsic localization of frames. Constr. Approx. 22(3), 395–415 (2005)

    MathSciNet  MATH  CrossRef  Google Scholar 

  18. F. Futamura, Banach framed, decay in the context of localization. Sampl. Theory Signal Image Process. 6(2), 151–166 (2007)

    MathSciNet  MATH  Google Scholar 

  19. T. Gramchev, S. Pilipović, L. Rodino, Classes of degenerate elliptic operators in Gelfand-Shilov spaces, in New Developments in Pseudo-Differential Operators. Operator Theory: Advances and Applications, vol. 189 (Birkhäuser, Basel, 2009), pp. 15–31

    MATH  CrossRef  Google Scholar 

  20. T. Gramchev, S. Pilipović, L. Rodino, Eigenfunction expansions in \(\mathbb {R}^n\). Proc. Am. Math. Soc. 139(12), 4361–4368 (2011)

    Google Scholar 

  21. K. Gröchenig, Describing functions: atomic decompositions versus frames. Monatsh. Math. 112(1), 1–42 (1991)

    MathSciNet  MATH  CrossRef  Google Scholar 

  22. K. Gröchenig, Localization of frames, Banach frames, and the invertibility of the frame operator. J. Fourier Anal. Appl. 10(2), 105–132 (2004)

    MathSciNet  MATH  CrossRef  Google Scholar 

  23. M. Guillemot-Teissier, Développements des distributions en séries de fonctions orthogonales. Séries de Legendre et de Laguerre (French). (Development of distributions in series of orthogonal functions. Series of Legendre an Laguerre). Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat. III. Ser. 25, 519–573 (1971)

    Google Scholar 

  24. S. Jakšić, S. Pilipović, B. Prangoski, G-type spaces of ultradistributions over \(\mathbb {R}^d_{+}\) and the Weyl pseudo-differential operators with radial symbols. Revista de la Real Academia de Ciencias Exactas, Fisicas y Naturales. Serie A. Matematicas 111(3), 613–640 (2017)

    Google Scholar 

  25. S. Pilipović, Tempered ultradistributions. Boll. Unione Mat. Ital. 7(2-B), 235–251 (1988)

    Google Scholar 

  26. S. Pilipović, D.T. Stoeva, Series expansions in Fréchet spaces and their duals, construction of Fréchet frames. J. Approx. Theory 163, 1729–1747 (2011)

    MathSciNet  MATH  CrossRef  Google Scholar 

  27. S. Pilipović, D.T. Stoeva, Fréchet frames, general definition and expansions. Anal. Appl. 12(2), 195–208 (2014)

    MathSciNet  MATH  CrossRef  Google Scholar 

  28. S. Pilipović, D.T. Stoeva, Localization of Fréchet frames and expansion of generalized functions (in preparation)

    Google Scholar 

  29. S. Pilipović, D.T. Stoeva, N. Teofanov, Frames for Fréchet spaces. Bull. Cl. Sci. Math. Nat. Sci. Math. 32, 69–84 (2007)

    MATH  Google Scholar 

  30. B. Simon, Distributions and their Hermite expansions. J. Math. Phys. 12(1), 140–148 (1971)

    MathSciNet  MATH  CrossRef  Google Scholar 

  31. D. Vuckovic, J. Vindas, Eigenfunction expansions of ultradifferentiable functions and ultradistributions in \(\mathbb {R}^n\). J. Pseudo-Differ. Oper. Appl. 7(4), 519–531 (2016)

    MathSciNet  MATH  CrossRef  Google Scholar 

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Acknowledgements

The authors acknowledge support from the Multilateral S&T Danube-Cooperation Project TIFMOFUS (“Time-Frequency Methods for Operators and Function Spaces”; MULT_DR 01/2017), the Austrian Science Fund (FWF) START-project FLAME (“Frames and Linear Operators for Acoustical Modeling and Parameter Estimation”; Y 551-N13), and the Project 174024 of the Serbian Ministry of Sciences. The second author is grateful for the hospitality of the University of Novi Sad, where most of the research on the presented topic was done. The authors express their gratitude to the referee for the valuable advices.

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Correspondence to Diana T. Stoeva .

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Pilipović, S., Stoeva, D.T. (2019). Frame Expansions of Test Functions, Tempered Distributions, and Ultradistributions. In: Lindahl, K., Lindström, T., Rodino, L., Toft, J., Wahlberg, P. (eds) Analysis, Probability, Applications, and Computation. Trends in Mathematics(). Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-04459-6_44

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