A functional model for the Fourier--Plancherel operator truncated on the positive half-axis

2017/10/30 by Victor Katsnelson, Katsnelson, Victor
Mathematics · #35S30 #47A45 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #math.CA #msc:35S30 #msc:47A45

paper · pdf · doi:10.48550/arxiv.1710.11494

17 pages. arXiv admin note: text overlap with arXiv:1208.3817

arxiv created 2018/07/17 · arxiv updated 2018/07/18

Abstract

The truncated Fourier operator \mathscrF_\mathbbR⁺, (\mathscrF_\mathbbR⁺x)(t)=(1)/(√(2π)) ∫_\mathbbR⁺x(ξ)e^itξ dξ , t∈\mathbbR⁺, is studied. The operator \mathscrF_\mathbbR⁺ is considered as an operator acting in the space L²(\mathbbR⁺). The functional model for the operator \mathscrF_\mathbbR⁺ is constructed. This functional model is the multiplication operator on the appropriate 2×2 matrix function acting in the space L²(\mathbbR⁺)⊕L²(\mathbbR⁺). Using this functional model, the spectrum of the operator \mathscrF_\mathbbR⁺ is found. The resolvent of the operator \mathscrF_\mathbbR⁺ is estimated near its spectrum.

Citations

• The spectral theory of the Fourier operator truncated on the positive half-axis

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