A functional model for the Fourier–Plancherel operator truncated to the positive semiaxis
Algebra i analiz, Tome 30 (2018) no. 3, pp. 93-111
Cet article a éte moissonné depuis la source Math-Net.Ru
The truncated Fourier operator $\mathscr F_{\mathbb R^+}$, \begin{equation*} (\mathscr F_{\mathbb R^+}x)(t)=\frac1{\sqrt{2\pi}}\int_{\mathbb R^+}x(\xi)e^{it\xi}\,d\xi,\quad t\in\mathbb{R^+}, \end{equation*} is studied. The operator $\mathscr F_{\mathbb R^+}$ is viewed as an operator acting in the space $L^2(\mathbb R^+)$. A functional model for the operator $\mathscr F_{\mathbb R^+}$ is constructed. This functional model is the operator of multiplication by an appropriate ($2\times2$)-matrix function acting in the space $L^2(\mathbb R^+)\oplus L^2(\mathbb R^+)$. Using this functional model, the spectrum of the operator $\mathscr F_{\mathbb R^+}$ is found. The resolvent of the operator $\mathscr F_{\mathbb R^+}$ is estimated near its spectrum.
Keywords:
truncated Fourier–Plancherel operator, functional model for a linear operator.
@article{AA_2018_30_3_a5,
author = {V. Katsnelson},
title = {A functional model for the {Fourier{\textendash}Plancherel} operator truncated to the positive semiaxis},
journal = {Algebra i analiz},
pages = {93--111},
year = {2018},
volume = {30},
number = {3},
language = {en},
url = {http://geodesic.mathdoc.fr/item/AA_2018_30_3_a5/}
}
V. Katsnelson. A functional model for the Fourier–Plancherel operator truncated to the positive semiaxis. Algebra i analiz, Tome 30 (2018) no. 3, pp. 93-111. http://geodesic.mathdoc.fr/item/AA_2018_30_3_a5/
[1] Sz.-Nagy B., Foias C., Bercovici H., Kérchy L., Harmonic analysis of operators on Hilbert space, Universitext, Springer, New York, 2010 | MR | Zbl
[2] Titchmarch E. C., Introduction to the theory of Fourier integrals, Third Ed., Chelsea Publ. Co., New York, 1986 | MR
[3] Bogachev V. I., Measure theory, v. 1, Springer-Verlag, Berlin, 2007 | MR | Zbl