A note on holomorphic functions and the Fourier-Laplace transform

  • Marcus Carlsson
  • Jens Wittsten

Abstract

We revisit the classical problem of when a given function, which is analytic in the upper half plane $\mathbb{C} _+$, can be written as the Fourier transform of a function or distribution with support on a half axis $(-\infty ,b]$, $b\in \mathbb{R} $. We derive slight improvements of the classical Paley-Wiener-Schwartz Theorem, as well as softer conditions for verifying membership in classical function spaces such as $H^p(\mathbb{C} _+)$.

References

Ben Cheikh, Y., Generalized Paley-Wiener theorem and generalized convolution associated with the differential operator $D^n_z$ in the complex domain, Integral Transform. Spec. Funct. 9 (2000), no. 4, 245–256. https://doi.org/10.1080/10652460008819259

>

Chung, J., Chung, S.-Y., and Kim, D., Characterizations of the Gelfand-Shilov spaces via Fourier transforms, Proc. Amer. Math. Soc. 124 (1996), no. 7, 2101–2108. https://doi.org/10.1090/S0002-9939-96-03291-1

>

Duren, P. L., Theory of $H^p$ spaces, Pure and Applied Mathematics, Vol. 38, Academic Press, New York-London, 1970, Dover Publications, 2000.

Ehrenpreis, L., Analytic functions and the Fourier transform of distributions. I, Ann. of Math. (2) 63 (1956), 129–159.

Folland, G. B., Real analysis: Modern techniques and their applications, second ed., Pure and Applied Mathematics (New York), John Wiley & Sons, New York, 1999.

Friedlander, F. G., Introduction to the theory of distributions, Cambridge University Press, Cambridge, 1982.

Gelfand, I. M. and Shilov, G. E., Generalized functions. Vol. 2. Spaces of fundamental and generalized functions, Translated from the Russian by Morris D. Friedman, Amiel Feinstein and Christian P. Peltzer, Academic Press, New York-London, 1968.

Hasumi, M., Note on the $n$-dimensional tempered ultra-distributions, Tôhoku Math. J. (2) 13 (1961), 94–104.

Hörmander, L., The analysis of linear partial differential operators. I: Distribution theory and Fourier analysis, Grundlehren der Mathematischen Wissenschaften, vol. 256, Springer-Verlag, Berlin, 1983. https://doi.org/10.1007/978-3-642-96750-4

>

Hörmander, L., The analysis of linear partial differential operators. III: Pseudodifferential operators, Grundlehren der Mathematischen Wissenschaften, vol. 274, Springer-Verlag, Berlin, 1985.

Koosis, P., The logarithmic integral. I, Cambridge Studies in Advanced Mathematics, vol. 12, Cambridge University Press, Cambridge, 1988. https://doi.org/10.1017/CBO9780511566196

>

Koosis, P., Introduction to $H_p$ spaces, second ed., Cambridge Tracts in Mathematics, vol. 115, Cambridge University Press, Cambridge, 1998.

Levin, B. J., Distribution of zeros of entire functions, revised ed., Translations of Mathematical Monographs, vol. 5, American Mathematical Society, Providence, R.I., 1980.

Martinez, A., An introduction to semiclassical and microlocal analysis, Universitext, Springer-Verlag, New York, 2002. https://doi.org/10.1007/978-1-4757-4495-8

>

Newman, D. J., An entire function bounded in every direction, Amer. Math. Monthly 83 (1976), no. 3, 192–193.

Ozaktas, H. M., Zalveeksy, Z., and Alper Kutay, M., The fractional Fourier transform: with applications in optics and signal processing, John Wiley & Sons, New York, 2001.

Paley, R. E. A. C. and Wiener, N., Fourier transforms in the complex domain, American Mathematical Society Colloquium Publications, vol. 19, American Mathematical Society, Providence, RI, 1987, Reprint of the 1934 original.

Rudin, W., Functional analysis, second ed., International Series in Pure and Applied Mathematics, McGraw-Hill, Inc., New York, 1991.

Schwartz, L., Transformation de Laplace des distributions, Comm. Sém. Math. Univ. Lund 1952 (1952), no. Tome Suppl., 196–206.

Sebastião e Silva, J., Les fonctions analytiques comme ultra-distributions dans le calcul opérationnel, Math. Ann. 136 (1958), 58–96.

Shambayati, R. and Zieleźny, Z., On Fourier transforms of distributions with one-sided bounded support, Proc. Amer. Math. Soc. 88 (1983), no. 2, 237–243. https://doi.org/10.2307/2044708

>

Trèves, F., Topological vector spaces, distributions and kernels, Academic Press, New York-London, 1967.

Vladimirov, V. S., Generalized functions in mathematical physics, “Mir”, Moscow, 1979.

Yoshinaga, K., On spaces of distributions of exponential growth, Bull. Kyushu Inst. Tech. Math. Nat. Sci. 6 (1960), 1–16.

Published
2017-05-27
How to Cite
Carlsson, M., & Wittsten, J. (2017). A note on holomorphic functions and the Fourier-Laplace transform. MATHEMATICA SCANDINAVICA, 120(2), 225-248. https://doi.org/10.7146/math.scand.a-25612
Section
Articles