Spectral and Tiling properties of the Unit Cube

By Alex Iosevich, S. Pedersen. 2001

77 citations recorded by the source corpus.

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.

Subfields: Computational Theory and Mathematics, Materials Chemistry

View Paper: https://www.semanticscholar.org/paper/7bedfcff2b4229960732119d8dbed0a0a58ef091

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Citations in the corpus

Fuglede’s conjecture fails in dimension 4 (2005, background): “To assess our present knowledge concerning this topic it is interesting to note that Fuglede’s conjecture was recently proved in 2 dimensions for convex bodies in [5].”
Fuglede’s conjecture for a union of two intervals (2000, background): “, n [2], [7], [16]; other relations between the tiling and spectral properties of subsets of Rn have been conjectured and in some cases proved, see [6], [8], [9], [11], [12], [14].”
Orthonormal bases of exponentials for the n-cube (2000, background): “1 Second, the tiling result in §3 applies to more general sets than linearly transformedn-cubes A = A([0,1]n); we give the onedimensional example = [0,1]∪[2,3].”
The spectral set conjecture and multiplicative properties of roots of polynomials (2000, unknown): “, [3], [4], [6], [8], [11], [9], [12], [13], [16], [17], [18], [19].”