The truncated Fourier operator. I

2009/01/16 by Victor Katsnelson, Katsnelson, Victor, Ronny Machluf +1
Mathematics · #47A10 #47A38 #47B06 #47B35 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Spectral Theory (math.SP) #math.CA #math.SP #msc:47A10 #msc:47A38 #msc:47B06 #msc:47B35

paper · pdf · doi:10.48550/arxiv.0901.2555

20 pages

arxiv created 2009/01/16 · arxiv updated 2009/12/01

Abstract

Let (\mathscrF) be the one dimensional Fourier-Plancherel operator and (E) be a subset of the real axis. The truncated Fourier operator is the operator (\mathscrF_E) of the form (\mathscrF_E=P_E\mathscrFP_E), where ((P_Ex)(t)=χ_E(t)x(t)), and (χ_E(t)) is the indicator function of the set (E). In the presented first part of the work, the basic properties of the operator (\mathscrF_E) according to the set (E) are discussed. Among these properties there are the following one. The operator (\mathscrF_E): 1. has a not-trivial null-space; 2. is strictly contractive; 3. is a normal operator; 4. is a Hilbert-Schmidt operator; 5. is a trace class operator.

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