Exponential spectra inL2(μ)

Frames of multi-windowed exponentials on subsets ofRd

“…(2) Another problem of a similar nature asks whether or not a Fourier frame exists on the singular one-third Cantor measure. In the existing methods, the construction of a Fourier frame is based on the existence of a singular measure for which there exists an orthonormal basis of exponentials [15,6]. While it is known that the one-third Cantor measure cannot admit any exponential orthogonal basis [17], we are interested in the existence of Fourier frames for a measure which genuinely cannot be derived from some already existing tight frames.…”

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].…”

“…We call a countable subset Λ ⊂ R d a spectrum of μ if the set of complex exponentials E Λ = {e λ : λ ∈ Λ} forms an orthonormal basis for L 2 (μ), where e λ (x) = e −2πi x,λ . If L 2 (μ) admits a spectrum, then μ is called a spectral measure; in this case, μ is of pure type, i.e., it is either discrete and finite, or absolutely continuous or singular continuous with respect to the Lebesgue measure [14]. In particular, if μ is the normalized Lebesgue measure supported on a Borel set Ω, then Ω is called a spectral set.…”

Spectrality of the planar Sierpinski family

“…It relates analysis, geometry and topology (see, e.g. [7,14,16,[26][27][28]30] and references therein), in which the good functions are complex exponentials. The best approximation appears when L 2 (μ) has a basis consisting of complex exponentials (Fourier basis).…”