Abstract
This article gives necessary conditions and slightly stronger sufficient conditions for a holomorphic function to be the Segal-Bargmann transform of a function inL p (ℝd, ρ) where ρ is a Gaussian measure. The proof relies on a family of inversion formulas for the Segal-Bargmann transform, which can be “tuned” to give the best estimates for a given value of p. This article also gives a single necessary-and-sufficient condition for a holomorphic function to be the transform of a function f such that any derivative of f multiplied by any polynomial is in Lp (d, ρ). Finally, I give some weaker but dimension-independent conditions.
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References
Baez, J., Segal, I., and Zhou, Z. (1992).Introduction to Algebraic and Constructive Quantum Field Theory, Princeton University Press, Princeton, NJ.
Bargmann, V. (1961). On a Hilbert space of analytic functions and an associated integral transform, Part I,Comm. Pure Appl. Math.,14, 187–214.
Bargmann, V. (1967). On a Hilbert space of analytic functions and an associated integral transform, Part II. A family of related function spaces. Application to distribution theory,Comm. Pure Appl. Math.,20, 1–101.
Davies, E.B., Gross, L., and Simon, B. (1992). Hypercontractivity: a bibliographic review, inIdeas and Methods in Quantum and Statistical Physics, (Oslo, 1988), Albeverio, S., et al., Eds., 370–389. Cambridge University Press, Cambridge.
Davies, E.B. and Simon, B. (1984). Ultracontractivity and the heat kernel for Schrödinger operators and Dirichlet Laplacians,J. Funct. Anal.,59, 335–395.
Driver, B. and Hall, B. (1999). Yang-Mills theory and the Segal-Bargmann transform.Commun. Math. Phys.,201, 249–290.
Feichtinger, H. and Stromer, T., Eds. (1998).Gabor Analysis and Algorithms, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Boston.
Folland, G. (1989).Harmonic Analysis in Phase Space, Princeton University Press, Princeton, NJ.
Glimm, J. and Jaffe, A. (1987).Quantum Physics. A Functional Integral Point of View, Second edition, Springer-Verlag, New York, Berlin.
Gross, L. (1975). Logarithmic Sobolev inequalities,Am. J. Math.,97, 1061–1083.
Gross, L. and Malliavin, P. (1996). Hall's transform and the Segal-Bargmann map, inItô's stochastic calculus and probability theory, Fukushima, M., Ikeda, N., Kunita, H., and Watanabe, S., Eds., 73–116. Springer-Verlag, Berlin/New York.
Hall, B. (1994). The Segal-Bargmann “coherent state” transform for compact Lie groups,J. Funct. Anal.,122, 103–151.
Hall, B. (1997). Phase space bounds for quantum mechanics on a compact Lie group,Comm. Math. Physics,184, 233–250.
Hall, B. (1999). A new form of the Segal-Bargmann transform for Lie groups of compact type.Canad. J. Math.,51, 816–834.
Hall, B. (2000). Holomorphic methods in analysis and mathematical physics, inFirst Summer School in Analysis and Mathematical Physics, Pérez-Esteva, S. and Villegas-Blas, C., Eds.,Contemp. Math., Vol. 260,Am. Math. Soc., Providence, RI, 1–59.
Hall, B. (2001). Harmonic analysis with respect to heat kernel measure,Bull. (N.S.) Am. Math. Soc.,38, 43–78.
Klauder, J. and Skagerstam, B. -S., Eds., (1985).Coherent States. Applications in Physics and Mathematical Physics, World Scientific Publishing Co., Singapore.
Nelson, E. (1973). The free Markoff field,J. Funct. Anal.,12, 211–227.
Perelomov, A. (1986).Generalized Coherent States and their Applications, Texts and Monographs and Physics, Springer-Verlag, Berlin-New York.
Rawnsley, J., Cahen, M., and Gutt, S. (1990). Quantization of Kähler manifolds, I. Geometric interpretation of Berezin's quantization,J. Geom. Phys.,7, 45–62.
Segal, I. (1978). The complex wave representation of the free Boson field, inTopics in functional analysis: Essays dedicated to M.G. Krein on the occasion of his 70th birthday, Gohberg, I. and Kac, M., Eds., Advances in Mathematics Supplementary Studies, Vol. 3, 321–343. academic Press, New York.
Sengupta, A. The two-parameter Segal-Bargmann transform, preprint (1999).
Sontz, S. (1998). Entropy and the Segal-Bargmann transform,J. Math. Phys.,39, 2402–2417.
Stein, E. and Weiss, G. (1971).Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, NJ.
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Communicated by Robert S. Strichartz
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Hall, B.C. Bounds on the Segal-Bargmann transform ofL p functions. The Journal of Fourier Analysis and Applications 7, 553–569 (2001). https://doi.org/10.1007/BF02513076
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DOI: https://doi.org/10.1007/BF02513076