Advertisement

Characterization of L. Schwartz’ convolutor and multiplier spaces \(\mathcal O _{C}'\) and \(\mathcal O _{M}\) by the short-time Fourier transform

  • Christian BargetzEmail author
  • Norbert Ortner
Original Paper

Abstract

A new definition of the short-time Fourier transform for temperate distributions is presented and its mapping properties are investigated. K.-H. Gröchenig and G. Zimmermann characterized the spaces \(\mathcal S \) and \(\mathcal S '\) of rapidly decreasing functions and temperate distributions, respectively, by their short-time Fourier transform. Following an idea of G. Zimmermann, we give analogous characterizations of the spaces \(\mathcal O _{C}'\) and \(\mathcal O _{M}\). These spaces, being (PLB)-spaces, have a much more complicated structure than \(\mathcal S \) and \(\mathcal S '\), which is the reason why we have to use the technical machinery of L. Schwartz’ theory of vector-valued distributions.

Keywords

Temperate and (very) rapidly decreasing distributions and functions Fourier transform Short-time Fourier transform 

Mathematics Subject Classification (2010)

Primary 42B10 Secondary 46F12 

Notes

Acknowledgments

We are indebted to Prof. G. Zimmermann who directed the second author’s attention to the mapping properties of the short-time Fourier transform and to their inversion. Preliminary versions of Propositions 4, 4., 5, 2., 7, 2., 10, 1., 2., and 11, 1., 2 are due to him.

References

  1. 1.
    Alexander, G.: Produits tensoriels topologiques et espaces nucléaires. Mem. Amer. Math. Soc. 1955(16):Chap I, 196, Chap II, 140 (1955)Google Scholar
  2. 2.
    Bierstedt, K.D.: An introduction to locally convex inductive limits. In: Functional Analysis and Its Applications (Nice, 1986), ICPAM Lecture Notes, pp. 35–133. World Sci. Publishing, Singapore (1988)Google Scholar
  3. 3.
    Christian, B.: Topological Tensor Products and the Convolution of Vector-Valued Distributions. PhD thesis, Universität Innsbruck (2012)Google Scholar
  4. 4.
    Folland, G.B.: Harmonic analysis in phase space, volume 122 of annals of mathematics studies. Princeton University Press, Princeton (1989)Google Scholar
  5. 5.
    Gröchenig, K., Zimmermann, G.: Hardy’s theorem and the short-time Fourier transform of Schwartz functions. J. London Math. Soc. (2) 63(1), 205–214 (2001)CrossRefzbMATHMathSciNetGoogle Scholar
  6. 6.
    Gröchenig, K., Zimmermann, G.: Spaces of test functions via the STFT. J. Funct. Spaces Appl. 2(1), 25–53 (2004)CrossRefzbMATHMathSciNetGoogle Scholar
  7. 7.
    Grothendieck, A.: Sur les espaces (\(F\)) et (\(DF\)). Summa Brasil. Math. 3, 57–123 (1954)MathSciNetGoogle Scholar
  8. 8.
    Horváth, J.: Topological Vector Spaces and Distributions. Addison-Wesley Publishing Co, Reading (1966)zbMATHGoogle Scholar
  9. 9.
    Jarchow, H.: Locally convex spaces. Mathematische Leitfäden. B. G. Teubner, Stuttgart (1981)CrossRefGoogle Scholar
  10. 10.
    Krantz, S.G., Parks, H.R.: A primer of real analytic functions. Birkhäuser Advanced Texts: Basler Lehrbücher, 2nd edn. Birkhäuser Boston Inc, Boston (2002)CrossRefGoogle Scholar
  11. 11.
    Laurent, S.: Espaces de fonctions différentiables à valeurs vectorielles. J. Analyse Math. 4:88–148, 1954–55Google Scholar
  12. 12.
    Laurent, S.: Théorie des noyaux. In: Proceedings of the International Congress of Mathematicians, vol 1, pp. 220–230. American Mathematical Society, Cambridge (1950)Google Scholar
  13. 13.
    Laurent, S.: Théorie des distributions. Publications de l’Institut de Mathématique de l’Université de Strasbourg, No. IX-X. Nouvelle édition, entièrement corrigée, refondue et augmentée. Hermann, Paris (1966)Google Scholar
  14. 14.
    Schwartz, L.: Lectures on Mixed Problems in Partial Differential Equations and Representations of Semi-groups. Tata Institute of Fundamental Research, Bombay (1957)Google Scholar
  15. 15.
    Schwartz, L.: Théorie des distributions à valeurs vectorielles. I. Ann. Inst. Fourier, Grenoble 7, 1–141 (1957) Google Scholar
  16. 16.
    Schwartz, L.: Théorie des distributions à valeurs vectorielles. II. Ann. Inst. Fourier. Grenoble 8, 1–209 (1958)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Vo-Khac, K.: Distributions, Analyse de Fourier, Opérateurs aux Derivées Partielles: cours et exercices résolus, avec une étude introductive des espaces vectoriels topologiques, Tome I. Vuibert, Paris (1972)Google Scholar

Copyright information

© Springer-Verlag Italia 2013

Authors and Affiliations

  1. 1.Institut für MathematikUniversität Innsbruck InnsbruckAustria

Personalised recommendations