Fourier bases and a distance problem of Erd\H os
By Alex Iosevich, Nets Hawk Katz, S. Pedersen. 2001
49 citations recorded by the source corpus.
We prove that no ball admits a non-harmonic orthogonal basis of exponentials. We use a combinatorial result, originally studied by Erd\H os, which says that the number of distances determined by $n$ points in ${\Bbb R}^d$ is at least $C_d n^{\frac{1}{d}+\epsilon_d}$, $\epsilon_d>0$.
Subfields: Applied Mathematics, Numerical Analysis
View Paper: https://www.semanticscholar.org/paper/bdf235e1164ac13eacbdc3ccbbcc88f2ab56b568
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In Applied Mathematics
- Dorin Ervin Dutkay
- Palle E. T. Jorgensen
- Izabella Laba
- Jian-Lin Li
- Xinggang He
- Xin-Rong Dai
- Chun-Kit Lai
- Ka-Sing Lau
In Numerical Analysis
Citations in the corpus
- Analysis of orthogonality and of orbits in affine iterated function systems (2006, methodology): “Since then they have been used extensively in harmonic analysis, in varied contexts involving the analysis of Fourier bases and geometry in for example the papers [Haa96, IoPe98, IoKaPe99 , PeWa01, LaWa96, LaWa97, Tao04] among others.”
- Fuglede’s conjecture for a union of two intervals (2000, background): “Recently there has been significant progress on the special case of the conjecture when is assumed to be convex [10], [ 3 ], [4], and in particular the 2-dimensional convex case appears to be nearly resolved [5].”
- Fuglede's spectral set conjecture for convex polytopes (2016, unknown): “It is known that the ball is not a spectral set [Iosevich et al. 1999; Fuglede 2001], nor any convex body with a smooth boundary [Iosevich et al. 2001].”