A Note on Operator Sampling and Fractional Fourier Transform

Sampling theory for operators motivated by the operator identification problem in communications engineering has been developed during the last few years 1–4 . In 4 , Hong and Pfander gave an operator version of Kramer’s Lemma see 4, Theorem 25 . But they did not give any explicit kernel satisfying the hypotheses in 4, Theorem 25 other than the Fourier kernel. In this paper, we present that the kernel of the fractional Fourier transform satisfies the hypotheses in 4, Theorem 25 . Therefore, we give a new applicability of Kramer’s method. The FRFT—a generalization of the Fourier transform FT —has received much attention in recent years due to its numerous applications, including signal processing, quantum physics, communications, and optics 5–7 . Hong and Pfander studied the sampling theorem on the operators which are bandlimited in the FT sense see 4 . In this paper, we generalize their results to bandlimited operators in the FRFT sense. For f ∈ L2 R , its FRFT is defined by