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 Algebra i Analiz: Year: Volume: Issue: Page: Find

 Algebra i Analiz, 2018, Volume 30, Issue 3, Pages 93–111 (Mi aa1597)

Research Papers

A functional model for the Fourier–Plancherel operator truncated to the positive semiaxis

V. Katsnelson

Department of Mathematics, The Weizmann Institute, 76100, Rehovot, Israel

Abstract: 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.

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Document Type: Article
Language: English

Citation: V. Katsnelson, “A functional model for the Fourier–Plancherel operator truncated to the positive semiaxis”, Algebra i Analiz, 30:3 (2018), 93–111

Citation in format AMSBIB
\Bibitem{Kat18} \by V.~Katsnelson \paper A functional model for the Fourier--Plancherel operator truncated to the positive semiaxis \jour Algebra i Analiz \yr 2018 \vol 30 \issue 3 \pages 93--111 \mathnet{http://mi.mathnet.ru/aa1597} \elib{http://elibrary.ru/item.asp?id=32855066}