CUBO A Mathematical Journal Vol.12, N° 03, (241–253). October 2010

 

On the Weyl Transform with Symbol in the Gel’fand-Shilov Space and its Dual Space

 

YASUYUKI OKA

Department of Mathematics, Sophia University 7-1 Kioicho, Chiyoda-ku, Tokyo 102-8554, Japan email: yasuyu-o@hoffman.cc.sophia.ac.jp

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ABSTRACT

In this paper, we claim two subjects. One is that the Weyl transform with symbol in the Gel’fand-Shilov space l ^r_r , r ≥ 1/2 , is a trace class operator. The other one is that the Weyl transform with symbol in the generalized function (l ^r_r )^1, r ≥ 1/2 , is a continuous linear transformation from the Gel’fand-Shilov space l ^r_r to (l ^r_r )¹. As r > 1, Z. Lozanov- Crvenkovic and D. Perišic have proved in [6] this result. Our second claim includes their result.

Key words and phrases: Weyl transform, Gel’fand-Shilov space, Fourier-Wigner transform, trace class operator, Schwartz’s kernel theorem.

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RESUMEN

En este artículo afirmamos dos asuntos. El primero es que la transformada de Weyl con símbolo en el espacio de Gel’fand-Shilov l ^r_r , r ≥ 1/2 , es un operador de clase trazo. El segundo asunto es que la transformación de Weyl con símbolo en las funciones generalizadas (l ^r_r )¹, r ≥ 1/2 , es una transformación lineal continua del espacio Gel’fand-Shilov l ^r_r to (l ^r_r )¹ . Como r > 1, Z. Lozanov-Crvenkovic y D. Perišic probaron en [6] este resultado. Nuestro resultado incluye su resultado.

Math. Subj. Class.: 46F05; 46F15; 81R15; 81S40.

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References

[1] CAPPIELLO, M., GRAMCHEV, M.T. AND RODINO, L., Gelfand-Shilov spaces, pseudodifferential operators and localization operators, in Modern Trends in Pseudo- Differential Operators, Editors: Toft, J., Wong, M.W. and Zhu, H., Birkhäuser, 297–312.

[2] CHUNG, S.-Y., KIM, D. AND LEE, E.G., Schwartz kernel theorem for the Fourier hyperfunctions, Tsukuba J. Math., Vol. 19, N.2 (1995), 377–385.

[3] DONG, C. AND MATSUZAWA, T., l -space of Gel’fand-Shilov and differential equations, Japan. J. Math. Vol. 19, N.2, (1994), 227–239.

[4] GEL’FAND, I.M. AND SHILOV, G.E., Generalized Functions Vol. 2, Academy of Sciences Moscow, U.S.S.R, 1958.

[5] GRÖCHENIG, K. AND ZIMMERMANN, G., Spaces of test functions via the STFT, J. Function Spaces Appl., 2 (2004), 25–53.

[6] LOZANOV-CRVENKOVIC , Z. AND PERIŠiC , D., Kernel theorem for the space of tempered ultradistributions, Integral Transforms and Special Functions, Vol. 18, N.10, October (2007), 699–713.

[7] NAGAMACHI, S. AND MUGIBAYASHI, N.,Hyperfunction quantum field theory, Commun. math. Phys., 46 (1976), 119–134.

[8] OKA, Y., N-representation for l and l′, Sophia Univ. Master’s Thesis, 2002.

[9] POOL, J.C.T., Mathematical aspects of the Weyl correspondence, J. Math. Phys. vol. 7, N.1, January (1966), 66–76.

[10] SIMON, B., Distributions and their Hermite expansions, J. Math. Phys. vol. 12, N.1 (1971), 140–148.

[11] SIMON, B., The Weyl transform and L^p functions on phase space, Proc. Amer. Math. Soc., 116 (1992), 1045–1047.

[12] TOFT, J., Continuity properties for modulation spaces with applications to pseudodifferential calculus, I, J. Funct. Anal., 207 (2), (2004), 399–429.

[13] VOROS, A., An algebra of pseudodifferential operators and the asymptotics of quantum mechanics, J. Funct. Anal., 29 (1978), 104–132.

[14] WEYL, H., The Theory of Groups and Quantum Mechanics, Dover, New York, 1950.        [ Links ]

[15] WONG, M.W., Weyl Transforms, Springer-Verlag, New York, Inc., 1998.        [ Links ]

[16] YOSHINO, K. AND OKA, Y., Asymptotic expansions of the solutions to the heat equations with hyperfunctions initial value, Commun. Korean Math. Soc., 23 (2008), N.4, 555–565.

[17] ZHANG, G.-Z., Theory of distributions of S type and pansions, Chinese Math. Acta., 4 (1963), 211–221.