Characterization of nuclearity for Beurling–Björck spaces
Authors:
Andreas Debrouwere, Lenny Neyt and Jasson Vindas
Journal:
Proc. Amer. Math. Soc. 148 (2020), 5171-5180
MSC (2010):
Primary 46E10, 46F05; Secondary 42B10, 46A11, 81S30
DOI:
https://doi.org/10.1090/proc/15227
Published electronically:
September 18, 2020
MathSciNet review:
4163830
Full-text PDF
Abstract | References | Similar Articles | Additional Information
Abstract: We characterize the nuclearity of the Beurling–Björck spaces $\mathcal {S}^{(\omega )}_{(\eta )}(\mathbb {R}^d)$ and $\mathcal {S}^{\{\omega \}}_{\{\eta \}}(\mathbb {R}^d)$ in terms of the defining weight functions $\omega$ and $\eta$.
- Klaus D. Bierstedt, An introduction to locally convex inductive limits, Functional analysis and its applications (Nice, 1986) ICPAM Lecture Notes, World Sci. Publishing, Singapore, 1988, pp. 35–133. MR 979516, DOI https://doi.org/10.1007/s13116-009-0018-2
- Göran Björck, Linear partial differential operators and generalized distributions, Ark. Mat. 6 (1966), 351–407 (1966). MR 203201, DOI https://doi.org/10.1007/BF02590963
- Chiara Boiti, David Jornet, and Alessandro Oliaro, The Gabor wave front set in spaces of ultradifferentiable functions, Monatsh. Math. 188 (2019), no. 2, 199–246. MR 3900031, DOI https://doi.org/10.1007/s00605-018-1242-3
- C. Boiti, D. Jornet, and A. Oliaro, About the nuclearity of $\mathcal {S}_{(M_p)}$ and $\mathcal {S}_{\omega }$. In: Boggiatto P. et al. (eds), Advances in microlocal and time-frequency analysis, pp. 121–129. Applied and Numerical Harmonic Analysis, Birkhäuser, Cham, 2020. DOI 10.1007/978-3-030-36138-9$\_$6
- C. Boiti, D. Jornet, A. Oliaro, and G. Schindl, Nuclearity of rapidly decreasing ultradifferentiable functions and time-frequency analysis, Collect. Math., in press, 2020. DOI 10.1007/s13348-020-00296-0
- R. W. Braun, R. Meise, and B. A. Taylor, Ultradifferentiable functions and Fourier analysis, Results Math. 17 (1990), no. 3-4, 206–237. MR 1052587, DOI https://doi.org/10.1007/BF03322459
- Soon-Yeong Chung, Dohan Kim, and Sungjin Lee, Characterization for Beurling-Björck space and Schwartz space, Proc. Amer. Math. Soc. 125 (1997), no. 11, 3229–3234. MR 1443817, DOI https://doi.org/10.1090/S0002-9939-97-04221-4
- I. M. Gel′fand and G. E. Shilov, Generalized functions. Vol. 3: Theory of differential equations, Academic Press, New York-London, 1967. Translated from the Russian by Meinhard E. Mayer. MR 0217416
- I. M. Gel′fand and G. E. Shilov, Generalized functions. Vol. 2. Spaces of fundamental and generalized functions, Academic Press, New York-London, 1968. Translated from the Russian by Morris D. Friedman, Amiel Feinstein and Christian P. Peltzer. MR 0230128
- Karlheinz Gröchenig and Georg Zimmermann, Spaces of test functions via the STFT, J. Funct. Spaces Appl. 2 (2004), no. 1, 25–53. MR 2027858, DOI https://doi.org/10.1155/2004/498627
- Morisuke Hasumi, Note on the $n$-dimensional tempered ultra-distributions, Tohoku Math. J. (2) 13 (1961), 94–104. MR 131759, DOI https://doi.org/10.2748/tmj/1178244354
- Victor Havin and Burglind Jöricke, The uncertainty principle in harmonic analysis, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 28, Springer-Verlag, Berlin, 1994. MR 1303780
- I. I. Hirschman Jr., On the behaviour of Fourier transforms at infinity and on quasi-analytic classes of functions, Amer. J. Math. 72 (1950), 200–213. MR 32816, DOI https://doi.org/10.2307/2372147
- Lars Hörmander, A uniqueness theorem of Beurling for Fourier transform pairs, Ark. Mat. 29 (1991), no. 2, 237–240. MR 1150375, DOI https://doi.org/10.1007/BF02384339
- N. Levinson, Restrictions imposed by certain functions on their Fourier transforms, Duke Math. J. 6 (1940), 722–731. MR 2662
- Hans-Joachim Petzsche, Die Nuklearität der Ultradistributionsräume und der Satz vom Kern. I, Manuscripta Math. 24 (1978), no. 2, 133–171 (German, with English summary). MR 492653, DOI https://doi.org/10.1007/BF01310050
- François Trèves, Topological vector spaces, distributions and kernels, Academic Press, New York-London, 1967. MR 0225131
Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 46E10, 46F05, 42B10, 46A11, 81S30
Retrieve articles in all journals with MSC (2010): 46E10, 46F05, 42B10, 46A11, 81S30
Additional Information
Andreas Debrouwere
Affiliation:
Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Krijgslaan 281, 9000 Gent, Belgium
MR Author ID:
1154620
Email:
andreas.debrouwere@UGent.be
Lenny Neyt
Affiliation:
Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Krijgslaan 281, 9000 Gent, Belgium
MR Author ID:
1337474
ORCID:
0000-0001-8116-1487
Email:
lenny.neyt@UGent.be
Jasson Vindas
Affiliation:
Department of Mathematics: Analysis, Logic and Discrete Mathematics, Ghent University, Krijgslaan 281, 9000 Gent, Belgium
MR Author ID:
795097
ORCID:
0000-0002-3789-8577
Email:
jasson.vindas@UGent.be
Keywords:
Beurling–Björck spaces,
nuclear spaces,
ultradifferentiable functions,
the short-time Fourier transform,
time-frequency analysis methods in functional analysis
Received by editor(s):
August 28, 2019
Received by editor(s) in revised form:
February 15, 2020
Published electronically:
September 18, 2020
Additional Notes:
The first author was supported by FWO-Vlaanderen through the postdoctoral grant 12T0519N
The second author gratefully acknowledges support by Ghent University through the BOF-grant 01J11615.
The third author was supported by Ghent University through the BOF-grants 01J11615 and 01J04017.
Communicated by:
Ariel Barton
Article copyright:
© Copyright 2020
American Mathematical Society