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Weighted (PLB)-spaces of ultradifferentiable functions and multiplier spaces

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Abstract

We study weighted (PLB)-spaces of ultradifferentiable functions defined via a weight function (in the sense of Braun, Meise and Taylor) and a weight system. We characterize when such spaces are ultrabornological in terms of the defining weight system. This generalizes Grothendieck’s classical result that the space \({\mathcal {O}}_M\) of slowly increasing smooth functions is ultrabornological to the context of ultradifferentiable functions. Furthermore, we determine the multiplier spaces of Gelfand-Shilov spaces and, by using the above result, characterize when such spaces are ultrabornological. In particular, we show that the multiplier space of the space of Fourier ultrahyperfunctions is ultrabornological, whereas the one of the space of Fourier hyperfunctions is not.

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References

  1. Agethen, S., Bierstedt, K.D., Bonet, J.: Projective limits of weighted (LB)-spaces of continuous functions. Arch. Math. 92, 384–398 (2009)

    Article  MathSciNet  Google Scholar 

  2. Bargetz, C., Ortner, N.: Characterization of L. Schwartz’ convolutor and multiplier spaces \({\cal{O}}^{\prime }_{C}\) and \({\cal{O}}_{M}\) by the short-time Fourier transform. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A. Math. RACSAM 108, 833–847 (2014)

  3. Bierstedt, K.D., Meise, R., Summers, W.H.: A projective description of weighted inductive limits. Trans. Amer. Math. Soc. 272, 107–160 (1982)

    Article  MathSciNet  Google Scholar 

  4. Bölcskei, H., Janssen, A.J.E.M.: Gabor frames, unimodularity, and window decay. J. Fourier Anal. Appl. 6(3), 255–276 (2000)

    Article  MathSciNet  Google Scholar 

  5. Bonet, J., Meise, R., Melikhov, S.N.: A comparison of two different ways to define classes of ultradifferentiable functions. Bull. Belg. Math. Soc. Simon Stevin 14, 425–444 (2007)

    Article  MathSciNet  Google Scholar 

  6. Braun, R.W., Meise, R., Taylor, B.A.: Ultradifferentiable functions and Fourier analysis. Results Math. 17, 206–237 (1990)

    Article  MathSciNet  Google Scholar 

  7. Christensen, O.: Pairs of dual Gabor frame generators with compact support and desired frequency localization. Appl. Comput. Harmon. Anal. 20, 403–410 (2006)

    Article  MathSciNet  Google Scholar 

  8. Debrouwere, A., Neyt, L., Vindas, J.: On the space of ultradistributions vanishing at infinity. Banach J. Math. Anal. 14, 915–934 (2020)

    Article  MathSciNet  Google Scholar 

  9. Debrouwere, A., Neyt, L., Vindas, J.: The nuclearity of Gelfand-Shilov spaces and kernel theorems. Collect. Math. 72, 203–227 (2021)

    Article  MathSciNet  Google Scholar 

  10. Debrouwere, A., Vindas, J.: On the non-triviality of certain spaces of analytic functions. Hyperfunctions and ultrahyperfunctions of fast growth. Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A. Math. RACSAM 112, 473–508 (2018)

  11. Debrouwere, A., Vindas, J.: On weighted inductive limits of spaces of ultradifferentiable functions and their duals. Math. Nachr. 292, 573–602 (2019)

    Article  MathSciNet  Google Scholar 

  12. Debrouwere, A., Vindas, J.: Topological properties of convolutor spaces via the short-time Fourier transform. Trans. Amer. Math. Soc. 374, 829–861 (2021)

    Article  MathSciNet  Google Scholar 

  13. Dimovski, P., Prangoski, B., Velinov, D.: Multipliers and convolutors in the space of tempered ultradistributions. Novi Sad J. Math. 44, 1–18 (2014)

    MathSciNet  MATH  Google Scholar 

  14. Domański, P.: Classical \((PLS)\)-spaces: spaces of distributions, real analytic functions and their relatives, pp. 51–70, in: Orlicz Centenary Volume, Banach Center Publications, Warszawa, (2004)

  15. Gel’fand, I.M., Shilov, G.E.: Generalized functions. Vol. 2: Spaces of fundamental and generalized functions. Academic Press, New York-London, (1968)

  16. Gröchenig, K.: Foundations of time-frequency analysis. Birkhäuser Boston Inc, Boston, MA (2001)

    Book  Google Scholar 

  17. Grothendieck, A.: Produits tensoriels topologiques et espaces nucléaires, Mem. Amer. Math. Soc. 16, (1955)

  18. Heinrich, T., Meise, R.: A support theorem for quasianalytic functionals. Math. Nachr. 280, 364–387 (2007)

    Article  MathSciNet  Google Scholar 

  19. Janssen, A.J.E.M.: Signal analytic proofs of two basic results on lattice expansions. Appl. Comp. Harmonic Anal. 1, 350–354 (1994)

    Article  MathSciNet  Google Scholar 

  20. Kawai, T.: On the theory of Fourier hyperfunctions and its applications to partial differential equations with constant coefficients. J. Fac. Sci. Univ. Tokyo, Sec. IA 17, 467–517 (1970)

    MathSciNet  MATH  Google Scholar 

  21. Kim, D., Kim, K.W., Lee, E.L.: Convolution and multiplication operators in Fourier hyperfunctions. Integral Transforms Spec. Funct. 17, 53–63 (2006)

    Article  MathSciNet  Google Scholar 

  22. Komatsu, H.: Ultradistributions I. Structure theorems and a characterization. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 20, 25–105 (1973)

    MathSciNet  MATH  Google Scholar 

  23. Larcher, J., Wengenroth, J.: A new proof for the bornologicity of the space of slowly increasing functions. Bull. Belg. Math. Soc. Simon Stevin 5, 887–894 (2014)

    MathSciNet  MATH  Google Scholar 

  24. Morimoto, M., Convolutors for ultrahyperfunctions, Internat. Sympos. Math. Problems in Theoret. Phys. (Kyoto,: Lecture Notes in Phys., vol. 39. Springer-Verlag 1975, 49–54 (1975)

  25. Schwartz, L.: Théorie des distributions. Hermann, Paris (1966)

    MATH  Google Scholar 

  26. Soloviev, M.A.: Inclusion theorems for the Moyal multiplier algebras of generalized Gelfand-Shilov spaces. Integr. Equ. Oper. Theory 93, 52 (2021). https://doi.org/10.1007/s00020-021-02664-2

    Article  MathSciNet  MATH  Google Scholar 

  27. Valdivia, M.: A representation of the space \({\cal{O}}_{M}\). Math. Z. 77, 463–478 (1981)

    Article  Google Scholar 

  28. Kostadinova, S., Pilipović, S., Saneva, K., Vindas, J.: The short-time Fourier transform of distributions of exponential type and Tauberian theorems for S-asymptotics. Filomat 30, 3047–3061 (2016)

    Article  MathSciNet  Google Scholar 

  29. Vogt, D.: On the functors \(\operatorname{Ext}^{1}(E, F)\) for Fréchet spaces. Studia Math. 85, 163–197 (1987)

    Article  MathSciNet  Google Scholar 

  30. Wengenroth, J.: Derived functors in functional analysis. Springer-Verlag, Berlin (2003)

    Book  Google Scholar 

  31. Zharinov, V.V., Fourier ultrahyperfunctions, (Russian) Izv. Akad. Nauk SSSR Ser. Mat. 44,: 533–570. English translation: Math. USSR-Izv. 16(1981), 479–511 (1980)

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Authors and Affiliations

  1. Department of Mathematics and Data Science, Vrije Universiteit Brussel, Pleinlaan 2, 1050, Brussels, Belgium

    Andreas Debrouwere

  2. Department of Mathematics, Analysis, Logic and Discrete Mathematics, Ghent University, Krijgslaan 281, 9000, Gent, Belgium

    Lenny Neyt

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  1. Andreas Debrouwere
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  2. Lenny Neyt
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Correspondence to Andreas Debrouwere.

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Communicated by Karlheinz Gröchenig.

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A. Debrouwere was supported by FWO-Vlaanderen through the postdoctoral Grant 12T0519N.

L. Neyt gratefully acknowledges support by FWO-Vlaanderen through the postdoctoral Grant 12ZG921N.

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Debrouwere, A., Neyt, L. Weighted (PLB)-spaces of ultradifferentiable functions and multiplier spaces. Monatsh Math 198, 31–60 (2022). https://doi.org/10.1007/s00605-021-01664-z

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