On the spectra of a Cantor measure

Geometry of spectral pairs

“…Since then a substantial literature with many applications has emerged, see e.g., [4,9,15,16,26,28,29] and references cited there. In [2] the authors present a related but different kind of Fourier bases.…”

Families of Spectral Sets for Bernoulli Convolutions

“…This general framework includes subjects such as universal tiling sets and the dual spectral set conjecture (see [11,22,25,27,35]). Starting with [16], questions about Fourier duality have received considerable attention with respect to pure harmonic analysis [5,8,9,24,29,30,33,34] and with respect to applications such as wavelets, sampling, algorithms, martingales, and substitution-dynamical systems [2][3][4]26].…”

“…P is a bounded operator because S and S * are both bounded operators, and P = S 2 . Therefore, (11) can be written…”

Spectrality of a class of infinite convolutions

“…In another direction, Jorgensen and Pedersen [16] made a head start to study the spectral property of the self-similar measures. Nowadays, there is a large literature on this topic [1][2][3][4][5][6][7]9,[12][13][14][15][16][19][20][21][23][24][25][26][28][29][30]. Among those, one of the best known results is that if ρ = 1/q for some integer q > 1, then μ ρ,{0,1} is a spectral measure if and only if q is an even integer [16].…”

“…If all (b k , D k ) are admissible, there are various methods to construct an infinite set Λ such that E Λ is an orthonormal set of L 2 (μ {b k }, {D k } ) [1,3,6,16,21,23,28,29]. In general the difficult part is to verify the completeness of E Λ .…”