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The norm of the Fourier transform on finite abelian groups. (English) Zbl 1202.42065
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.

MSC:
42C40Wavelets and other special systems
43A25Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups
43A15$L^p$-spaces and other function spaces on groups, semigroups, etc.
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