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Let R R be an expanding matrix with integer entries, and let B , L B,L be finite integer digit sets so that ( R , B , L ) (R,B,L) form a Hadamard triple on R d {\mathbb {R}}^d in the sense that the matrix 1 | det R | [ e 2 π i ⟨ R − 1 b , ℓ ⟩ ] ℓ ∈ L , b ∈ B \begin{equation*} \frac {1}{\sqrt {|\det R|}}\left [e^{2\pi i \langle R^{-1}b,\ell \rangle }\right ]_{\ell \in L,b\in B} \end{equation*} is unitary. We prove that the associated fractal self-affine measure μ = μ ( R , B ) \mu = \mu (R,B) obtained by an infinite convolution of atomic measures μ ( R , B ) = δ R − 1 B ∗ δ R − 2 B ∗ δ R − 3 B ∗ ⋯ \begin{equation*} \mu (R,B) = \delta _{R^{-1} B}\ast \delta _{R^{-2}B}\ast \delta _{R^{-3}B}\ast \cdots \end{equation*} is a spectral measure, i.e., it admits an orthonormal basis of exponential functions in L 2 ( μ ) L^2(\mu ) . This settles a long-standing conjecture proposed by Jorgensen and Pedersen and studied by many other authors. Moreover, we also show that if we relax the Hadamard triple condition to an almost-Parseval-frame condition, then we obtain a sufficient condition for a self-affine measure to admit Fourier frames.

The parasympathetic activity before falling asleep and the sympathetic activity before waking up change coincidentally with EEG frequency, and may respectively contain the messages of sleeping and waking drives.

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