A functional model for the Fourier-Plancherel operator truncated to the positive semiaxis. (English) Zbl 1486.47075
St. Petersbg. Math. J. 30, No. 3, 457-469 (2019) and Algebra Anal. 30, No. 3, 93-111 (2018).
Summary: The truncated Fourier operator \(\mathcal {F}_{\mathbb{R}^+}\), \[(\mathcal {F}_{\mathbb{R}^+}x)(t)=\frac {1}{\sqrt {2\pi }}\int _{\mathbb{R}^+}x(\xi )e^{it\xi }\,d\xi ,\quad t\in {\mathbb{R}^+},\] is studied. The operator \(\mathcal {F}_{\mathbb{R}^+}\) is viewed as an operator acting in the space \(L^2(\mathbb{R}^+)\). A functional model for the operator \(\mathcal {F}_{\mathbb{R}^+}\) is constructed. This functional model is the operator of multiplication by an appropriate (\({2\times 2}\))-matrix function acting in the space \(L^2(\mathbb{R}^+)\oplus L^2(\mathbb{R}^+)\). Using this functional model, the spectrum of the operator \(\mathcal {F}_{\mathbb{R}^+}\) is found. The resolvent of the operator \(\mathcal {F}_{\mathbb{R}^+}\) is estimated near its spectrum.
MSC:
47B90 | Operator theory and harmonic analysis |
47B92 | Operators on real function spaces |
42A38 | Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type |
47A10 | Spectrum, resolvent |
References:
[1] | Sz B. Sz.-Nagy, C. Foias, H. Bercovici, and L. K\'erchy, Harmonic analysis of operators on Hilbert space, Universitext, Springer, New York, 2010. · Zbl 1234.47001 |
[2] | T E. C. Titchmarch, Introduction to the theory of Fourier integrals, Third ed., Chelsea Publ. Co., New York, 1986. |
[3] | Bo V. I. Bogachev, Measure theory. Vol. 1, Springer-Verlag, Berlin, 2007. · Zbl 1120.28001 |
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