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corrected misspelled name

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edit approved Apr 29, 2015 at 21:50

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Such T and measure do exist and in certain cases the measure is the restriction of usual Lebeasgue measure in $\mathbb{R}^n$. This kinds of sets are called spectral sets and spectral measures respectively, i.e., sets/ measures for which an complete orthogonal system of exponentials exist. The study of such sets began with the work of Fuglede (Fuglede, B. "Commuting Self-Adjoint Partial Differential Operators and a Group Theoretic Problem." J. Func. Anal. 16, 101-121, 1974.) who conjectured that a set is spectral if and only if it tiles and proved it in the special case when the spectrum or the tiling set is a lattice. The conjecture however has been disproved in dimension bigger than 3 following the work of TaoTerence Tao, T. "Fuglede's Conjecture Is False in 5 and Higher Dimensions." ( http://arxiv.org/abs/math.CO/0306134.) but remain open in dimension 1,2. Further under additional hypothesis on the set T many positive results are known. The study of spectral measure began with the work of Jorgensen and Pedersen and Strichartz.

Such T and measure do exist and in certain cases the measure is the restriction of usual Lebeasgue measure in $\mathbb{R}^n$. This kinds of sets are called spectral sets and spectral measures respectively, i.e., sets/ measures for which an complete orthogonal system of exponentials exist. The study of such sets began with the work of Fuglede (Fuglede, B. "Commuting Self-Adjoint Partial Differential Operators and a Group Theoretic Problem." J. Func. Anal. 16, 101-121, 1974.) who conjectured that a set is spectral if and only if it tiles and proved it in the special case when the spectrum or the tiling set is a lattice. The conjecture however has been disproved in dimension bigger than 3 following the work of Tao Tao, T. "Fuglede's Conjecture Is False in 5 and Higher Dimensions." ( http://arxiv.org/abs/math.CO/0306134.) but remain open in dimension 1,2. Further under additional hypothesis on the set T many positive results are known. The study of spectral measure began with the work of Jorgensen and Pedersen and Strichartz.

Such T and measure do exist and in certain cases the measure is the restriction of usual Lebeasgue measure in $\mathbb{R}^n$. This kinds of sets are called spectral sets and spectral measures respectively, i.e., sets/ measures for which an complete orthogonal system of exponentials exist. The study of such sets began with the work of Fuglede (Fuglede, B. "Commuting Self-Adjoint Partial Differential Operators and a Group Theoretic Problem." J. Func. Anal. 16, 101-121, 1974.) who conjectured that a set is spectral if and only if it tiles and proved it in the special case when the spectrum or the tiling set is a lattice. The conjecture however has been disproved in dimension bigger than 3 following the work of Terence Tao, "Fuglede's Conjecture Is False in 5 and Higher Dimensions." ( http://arxiv.org/abs/math.CO/0306134.) but remain open in dimension 1,2. Further under additional hypothesis on the set T many positive results are known. The study of spectral measure began with the work of Jorgensen and Pedersen and Strichartz.

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created Nov 12, 2011 at 13:16

Vagabond

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Such T and measure do exist and in certain cases the measure is the restriction of usual Lebeasgue measure in $\mathbb{R}^n$. This kinds of sets are called spectral sets and spectral measures respectively, i.e., sets/ measures for which an complete orthogonal system of exponentials exist. The study of such sets began with the work of Fuglede (Fuglede, B. "Commuting Self-Adjoint Partial Differential Operators and a Group Theoretic Problem." J. Func. Anal. 16, 101-121, 1974.) who conjectured that a set is spectral if and only if it tiles and proved it in the special case when the spectrum or the tiling set is a lattice. The conjecture however has been disproved in dimension bigger than 3 following the work of Tao Tao, T. "Fuglede's Conjecture Is False in 5 and Higher Dimensions." ( http://arxiv.org/abs/math.CO/0306134.) but remain open in dimension 1,2. Further under additional hypothesis on the set T many positive results are known. The study of spectral measure began with the work of Jorgensen and Pedersen and Strichartz.