Spectral and Tiling properties of the Unit Cube

Details Spectral and Tiling properties of the Unit Cube Spectral and Tiling properties of the Unit Cube

2001-04-08

arxiv.org/abs/math/0104093v1

International Math Research Notices, (1998), Number 16, pp. 819-828

Mathematics

Classical Analysis and ODEs

Scientific paper

Let $\Q=[0,1)^d$ denote the unit cube in $d$-dimensional Euclidean space \Rd
and let \T be a discrete subset of \Rd. We show that the exponentials
$e_t(x):=exp(i2\pi tx)$, $t\in\T$ form an othonormal basis for $L^2(\Q)$ if and
only if the translates $\Q+t$, $t\in\T$ form a tiling of \Rd.

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