On the Fourier transform of Lorentz invariant distributions. (English) Zbl 1227.46032
In the paper, formulas are given for the Fourier transforms of \(\delta_s([x,x])\) and \(f([x,x])\), where \(s\in\mathbb{R}\), \([x,x]= x^2_0- x^2_1-\cdots- x^2_{n-1}\); further, \(f\) is a locally integrable function satisfying certain conditions or it is the boundary value of a meromorphic function defined in the complex upper half-plane. The formulas are applied to yield tempered fundamental solutions of the iterated Klein-Gordon operator
\[ ([\partial, \partial]- c^2)^m,\;m\in\mathbb N,\quad c^2\in\mathbb{C}\setminus\mathbb{R}. \]
\[ ([\partial, \partial]- c^2)^m,\;m\in\mathbb N,\quad c^2\in\mathbb{C}\setminus\mathbb{R}. \]
Reviewer: László Simon (Budapest)
MSC:
| 46F12 | Integral transforms in distribution spaces |
| 35E05 | Fundamental solutions to PDEs and systems of PDEs with constant coefficients |
| 35L15 | Initial value problems for second-order hyperbolic equations |
| 74H05 | Explicit solutions of dynamical problems in solid mechanics |
Keywords:
Lorentz invariant distributions; Fourier transform; Klein-Gordon equation; tempered distributionReferences:
| [1] | M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions , 9th printing, Dover, New York, 1970. · Zbl 0543.33001 |
| [2] | N. N. Bogolubov, A. A. Logunov and I. T. Todorov, Axiomatic Quantum Field Theory , Benjamin, Reading, MA, 1975. |
| [3] | G. Friedlander and M. Joshi, Introduction to the Theory of Distributions , 2nd ed., Cambridge Univ. Press, Cambridge, 1998. · Zbl 0499.46020 |
| [4] | L. Gå rding and J.-L. Lions, Functional analysis , Nuovo Cimento 10 (1959), Suppl. 14, 9-66. · Zbl 0091.27602 · doi:10.1007/BF02724838 |
| [5] | I. M. Gel’fand and G. E. Shilov, Generalized Functions. Vol. I , Academic Press, New York, 1964. · Zbl 0115.33101 |
| [6] | I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series and Products , Academic Press, New York, 1980. · Zbl 0521.33001 |
| [7] | L. Hörmander, The Analysis of Linear Partial Differential Operators. Vol. I , Grundlehren Math. Wiss. 256 , Springer, Berlin, 1983. |
| [8] | L. Hörmander, The Analysis of Linear Partial Differential Operators. Vol. II , Grundlehren Math. Wiss. 257 , Springer, Berlin, 1983. |
| [9] | J. Horváth, Distribuciones definidas por prolongación analí tica , Rev. Colombiana Mat. 8 (1974), 7-95. |
| [10] | E. M. de Jager, Applications of Distributions in Mathematical Physics , Math. Centrum, Amsterdam, 1969. · Zbl 0195.27402 |
| [11] | A. I. Komech, Linear partial differential equations with constant coefficients , In: Partial Differential Equations. Vol. II (Enc. Math. Sci. Vol. 31, ed. by Yu.V. Egorov and M.A. Shubin), 121-255, Springer, Berlin, 1994. · Zbl 0805.35001 |
| [12] | J. Lavoine, Solution de l’équation de Klein-Gordon , Bull. Sci. Math. 85 (2) (1961), 57-72. · Zbl 0101.07702 |
| [13] | J. Leray, Hyperbolic Differential Equations , Institute of Advanced Study, Princeton, 1952. |
| [14] | P.-D. Methée, Sur les distributions invariantes dans le groupe des rotations de Lorentz , Commentarii Math. Helv. 28 (1954), 1-49. · Zbl 0055.34101 · doi:10.1007/BF02566932 |
| [15] | F. Oberhettinger, Tables of Bessel Transforms , Springer, Berlin, 1972. · Zbl 0261.65003 |
| [16] | F. Oberhettinger, Tables of Fourier Transforms and Fourier Transforms of Distributions , Springer, Berlin, 1990. · Zbl 0694.42017 |
| [17] | N. Ortner and P. Wagner, Distribution-Valued Analytic Functions. Theory and Applications , Lecture note 37 , Max-Planck-Institut, Leipzig, 2008; http://www.mis.mpg.de/preprints/ln/lecturenote-3708.pdf. |
| [18] | M. Riesz, L’intégrale de Riemann-Liouville et le problème de Cauchy , Acta Math. 81 (1949), 1-223; Collected papers: 571-793, Springer, Berlin, 1988. · Zbl 0033.27601 · doi:10.1007/BF02395016 |
| [19] | L. Schwartz, Sur les multiplicateurs de \(FL^p\!\!\) , Kungl. Fysiografiska Sällskapets i Lund Förhandlingar 22 (1953), 124-128. · Zbl 0047.35401 |
| [20] | L. Schwartz, Matemática y Fí sica Cuántica , Notas, Universidad de Buenos Aires, 1958. |
| [21] | L. Schwartz, Application of distributions to the theory of elementary particles in quantum mechanics , Gordon and Breach, New York, 1968. |
| [22] | L. Schwartz, Théorie des Distributions , Nouv. éd, Hermann, Paris, 1966. |
| [23] | R. S. Strichartz, Fourier transforms and non-compact rotation groups , Indiana Univ. Math. J. 24 (1974/75), 499-526. · Zbl 0295.42015 · doi:10.1512/iumj.1974.24.24037 |
| [24] | Z. Szmydt, Fourier Transformation and Linear Differential Equations , Reidel, Dordrecht, 1977. |
| [25] | S. E. Trione, On the Fourier transforms of retarded Lorentz-invariant functions , J. Math. Anal. Appl. 84 (1981), 73-112. · Zbl 0495.42012 · doi:10.1016/0022-247X(81)90153-0 |
| [26] | V. S. Vladimirov, Generalized Functions in Mathematical Physics , 2nd ed., Mir, Moscow, 1979. |
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