Abstract
We consider the counter images \(\mathcal H_\flat (\mathbf R^{d})\) and \(\mathcal H_{0,\flat }(\mathbf R^{d})\) of entire functions with exponential and almost exponential bounds, respectively, under the Bargmann transform, and we characterize them by estimates of powers of the harmonic oscillator. We also consider the Pilipović spaces \(\varvec{\mathcal S}_{\! s}(\mathbf R^{d})\) and \(\varvec{\varSigma }_s(\mathbf R^{d})\) when \(0<s<1/2\) and deduce their images under the Bargmann transform.
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References
Berend, D., Tassa, T.: Improved bounds on Bell numbers and on moments of sums of random variables. Probab. Math. Stat. 30, 185–205 (2010)
Chung, J., Chung, S.-Y., Kim, D.: Characterizations of the Gelfand–Shilov spaces via Fourier transforms. Proc. Am. Math. Soc. 124, 2101–2108 (1996)
de Bruijn, N.G.: Asymptotic Methods in Analysis. Bibliotheca Mathematica, 2nd edn. North-Holland Publishing Co., P. Noordhoff Ltd., Groningen (1961)
Gelfand, I.M., Shilov, G.E.: Generalized Functions, II–III. Academic Press, NewYork, London (1968)
Gramchev, T., Pilipović, S., Rodino, L.: Classes of degenerate elliptic operators in Gelfand–Shilov spaces. In: Rodino, L., Wong, M.W. (eds.) New Developments in Pseudo-Differential operators, Operator Theory: Advances and Applications, vol. 189, pp. 15–31. Birkhäuser, Basel (2009)
Gramchev, T., Pilipović, S., Rodino, L.: Eigenfunction expansions in \({\mathbb{R}}^n\). Proc. Am. Math. Soc. 139, 4361–4368 (2011)
Janssen, A.J.E.M., van Eijndhoven, S.J.L.: Spaces of type \(W,\) growth of Hermite coefficients, Wigner distribution, and Bargmann transform. J. Math. Anal. Appl. 152, 368–390 (1990)
Pilipović, S.: Generalization of Zemanian spaces of generalized functions which have orthonormal series expansions. SIAM J. Math. Anal. 17, 477–484 (1986)
Pilipović, S.: Tempered ultradistributions. Boll. UMI 7, 235–251 (1988)
Toft, J.: The Bargmann transform on modulation and Gelfand–Shilov spaces, with applications to Toeplitz and pseudo-differential operators. J. Pseudo-Differ. Oper. Appl. 3, 145–227 (2012)
Toft, J.: Images of function and distribution spaces under the Bargmann transform. arXiv:1409.5238 (preprint)
Vučović, D., Vindas, J.: Eigenfunction expansions of ultradifferentiable functions and ultradistributions in \({\mathbb{R}}^n\). arXiv:1512.01684
Zhang, G.Z.: Theory of distributions of \({\cal S}\) type and expansions. Acta Math. Sinica 13, 211–221 (1963) [(Chinese; translation in Chinese Math. Acta. 4, 211-221 (1963)]
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The research of C. Fernández and A. Galbis was partially supported by the projects MTM2013-43540-P, GVA Prometeo II/2013/013 and ACOMP/2015/186.
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Fernández, C., Galbis, A. & Toft, J. The Bargmann transform and powers of harmonic oscillator on Gelfand–Shilov subspaces. RACSAM 111, 1–13 (2017). https://doi.org/10.1007/s13398-015-0273-z
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DOI: https://doi.org/10.1007/s13398-015-0273-z