Frame multiplication theory and a vector-valued DFT and ambiguity function (Q2003615)

The authors establish the theory of vector-valued ambiguity functions of vector-valued functions $v$ on $\mathbb R^d$ in terms of their discrete periodic counterparts on $\mathbb Z/N\mathbb Z$. To this end they define the vector-valued discrete Fourier transforms (DFT) and the discrete periodic vector-valued ambiguity functions on $\mathbb Z/N\mathbb Z$ in a natural way. The outline of the paper is the following. Section 4 establishes the basic role of the short time Fourier transform (STFT) in achieving the above mentioned goals. In the process, the notion of frame multiplication is used to define the vector-valued ambiguity function. The theory of frame multiplication is formally presented in Section 5. In Section 6 the authors define the harmonic and group frames that are the basis for Abelian group frame multiplication results of Section 7. Section 8.2 is devoted to the uncertainty principle in the context of vector-valued DFT theory.