The norm of the Fourier transform on finite abelian groups (Q1958975)

Let \(G\) be a finite abelian group and \(\hat G\) its dual group. The Fourier transform on \(G\) is defined by \({\mathcal F}v(\gamma)=|G|^{\frac{1}{2}}\langle v,\gamma\rangle\) for \(\gamma\in\hat G\), where \(\langle\cdot,\cdot\rangle\) is the inner product on \({\mathbb C}^{|G|}\), and its norm is given by \(C_{p,q}=\sup_{\|v\|_p=1}\|{\mathcal F}v\|_q\) for \(1\leq p,q\leq\infty\). In this paper, the authors calculate the values of \(C_{p,q}\) and find the functions which attain the upper bound. They split the square \([0,1]^2=\{(\frac{1}{p},\frac{1}{q})\mid 1\leq p,q\leq\infty\}\) into three regions. In each region, an upper bound follows from Riesz-Thorin convexity theorem, and then it is attained by characters, delta functions, and biunimodular functions, respectively. Here, characters form a frequency basis on \(\hat G\), delta functions do a time basis on \(G\), and biunimodular functions are constructed from a time-frequency basis. They also characterize the set of extremals \(E_{p,q}=\{v\in L^p(G)\mid \|{\mathcal F}v\|_q=C_{p,q}\|v\|_p\}\). Especially, biunimodular functions appear in the above rigion and the so-called wave packets does in the classical range of \(p,q\) that corresponds to the Hausdorff-Young inequality on the real line. A survey on biunimodular functions is given.