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An uncertainty inequality for finite Abelian groups. (English) Zbl 1145.43005
Summary: Let $G$ be a finite Abelian group of order $n$. For a complex valued function $f$ on $G$ let $\widehat f$ denote the Fourier transform of $f$. The classical uncertainty inequality asserts that if $f\ne 0$ then $$|\text{supp}(f)|\cdot|\text{supp}(\widehat f)|\ge|G|.\tag 1$$ Answering a question of Terence Tao, the following improvement of (1) is shown: Theorem. Let $d_1<d_2$ be two consecutive divisors of $n$. If $d_1\le k=|\text{supp}(f)|\le d_2$, then $$|\text{supp}(\widehat f)|\ge\frac{n}{d_1d_2}(d_1+d_2-k).$$

MSC:
43A32Other transforms and operators of Fourier type
20K01Finite abelian groups
20C15Ordinary representations and characters of groups
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References:
[1] Donoho, D. L.; Stark, P. B.: Uncertainty principles and signal recovery. SIAM J. Appl. math. 49, 906-931 (1989) · Zbl 0689.42001
[2] Meshulam, R.: An uncertainty inequality for groups of order pq. European J. Combin. 13, 401-407 (1992) · Zbl 0790.43007
[3] Smith, K. T.: The uncertainty principle on groups. SIAM J. Appl. math. 50, 876-882 (1989)
[4] Tao, T.: An uncertainty principle for cyclic groups of prime order · Zbl 1080.42002
[5] Terras, A.: Fourier analysis on finite groups and applications. (1999) · Zbl 0928.43001