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A Paley–Wiener theorem in extended Gevrey regularity

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Abstract

In this paper we introduce appropriate associated function to the sequence \(M_p=p^{\tau p^{\sigma }}, p\in {\mathbf {N}}, \tau>0, \sigma >1\), and derive its sharp asymptotic estimates in terms of the Lambert W function. These estimates are used to prove a Paley–Wiener type theorem for compactly supported functions from extended Gevrey classes. As an application, we discuss properties of the corresponding wave front sets.

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References

  1. Cicognani, M., Lorentz, D.: Strictly hyperbolic equations with coefficients low-regular win time and smooth in space. J. Pseudo-Differ. Oper. Appl. 9, 643–675 (2018)

    Article  MathSciNet  Google Scholar 

  2. Corless, R.M., Gonnet, G.H., Hare, D.E.G., Jeffrey, D.J., Knuth, D.E.: On the Lambert W function. Adv. Comput. Math. 5, 329–359 (1996)

    Article  MathSciNet  Google Scholar 

  3. Gelfand, I.M., Shilov, G.E.: Generalized Functions II. Academic Press, New York (1968)

    Google Scholar 

  4. Hoorfar, A., Hassani, M.: Inequalities on the Lambert W function and hyperpower function. J. Inequal. Pure Appl. Math. 9, Article 51 (2008)

  5. Hörmander, L.: The Analysis of Linear Partial Differential Operators. Vol. I: Distribution Theory and Fourier Analysis. Springer, Berlin (1983)

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

    MathSciNet  Google Scholar 

  7. Komatsu, H.: An Introduction to the Theory of Generalized Functions. Lecture notes, Department of Mathematics Science University of Tokyo (1999)

  8. Pilipović, S., Teofanov, N., Tomić, F.: On a class of ultradifferentiable functions. Novi Sad J. Math. 45, 125–142 (2015)

    MathSciNet  MATH  Google Scholar 

  9. Pilipović, S., Teofanov, N., Tomić, F.: Beyond Gevrey regularity. J. Pseudo-Differ. Oper. Appl. 7, 113–140 (2016)

    Article  MathSciNet  Google Scholar 

  10. Pilipović, S., Teofanov, N., Tomić, F.: Superposition and propagation of singularities for extended Gevrey regularity. FILOMAT 32(8), 2763–2782 (2018)

    Article  MathSciNet  Google Scholar 

  11. Pilipović, S., Teofanov, N., Tomić, F.: Regulatiries for a new class of spaces between distributions and ultradistributions. Sarajevo J. Math. 14(2), 251–264 (2018)

    MathSciNet  MATH  Google Scholar 

  12. Rodino, L.: Linear Partial Differential Operators in Gevrey Spaces. World Scientific, Singapore (1993)

    Book  Google Scholar 

  13. Strichartz, R.S.: A Guide to Distribution Theory and Fourier Transforms. World Scientific, Singapore (2003)

    Book  Google Scholar 

  14. Teofanov, N., Tomić, F.: Inverse closedness and singular support in extended Gevrey regularity. J. Pseudo-Differ. Oper. Appl. 8, 411–421 (2017)

    Article  MathSciNet  Google Scholar 

  15. Teofanov, N., Tomić, F.: Ultradifferentiable functions of class \(M^{\tau ,\sigma } _p\) and microlocal regularity. In: Oberguggenberger, M., Toft, J., Vindas, J., Wahlberg, P. (eds.) Generalized Functions and Fourier Analysis, in Operator Theory: Advances and Applications, 260, Advanced Partial Differential Equations, pp. 193–213. Birkhäuser (2017)

  16. Tomić, F.: A microlocal property of PDOs in \({\cal{E}}_{(\tau ,\sigma )}(U)\). In: Proceedings of the Second Conference on Mathematics in Engineering: Theory and Applications, Novi Sad (2017)

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Acknowledgements

This research is supported by Ministry of Education, Science and Technological Development of Serbia through the Project No. 174024.

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Authors and Affiliations

  1. Department of Mathematics and Informatics, University of Novi Sad, Novi Sad, Serbia

    Stevan Pilipović & Nenad Teofanov

  2. Faculty of Technical Sciences, University of Novi Sad, Novi Sad, Serbia

    Filip Tomić

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  1. Stevan Pilipović
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  2. Nenad Teofanov
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  3. Filip Tomić
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Correspondence to Nenad Teofanov.

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Pilipović, S., Teofanov, N. & Tomić, F. A Paley–Wiener theorem in extended Gevrey regularity. J. Pseudo-Differ. Oper. Appl. 11, 593–612 (2020). https://doi.org/10.1007/s11868-019-00298-y

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  • DOI: https://doi.org/10.1007/s11868-019-00298-y

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