The truncated Fourier operator. General results (Q2912581)

The authors define the truncated Fourier operator by NEWLINE\[NEWLINE\mathcal F_E x(t)=\frac{1}{\sqrt{2\pi}}\int_Ee^{it\xi} x(\xi)\, d\xi,\quad t\in E, NEWLINE\]NEWLINE as an operator acting on functions in \(L^2(E)\), where \(E\) is a measurable set of the real axis with \(m(E)>0\). Several basic questions, depending on the set \(E\), of such an operator are studied, for instance, properties such as when the operator has non-trivial null-space, is strictly contractive, is normal, is Hilbert-Schmidt or is a trace class operator are analyzed.