Spectra of Bernoulli Convolutions as Multipliers in L on the Circle
نویسندگان
چکیده
It is shown that the closure of the set of Fourier coefficients of the Bernoulli convolution μθ parameterized by a Pisot number θ, is countable. Combined with results of Salem and Sarnak, this proves that for every fixed θ > 1 the spectrum of the convolution operator f 7→ μθ ∗ f in L(S) (where S is the circle group) is countable and is the same for all p ∈ (1,∞), namely, {μ̂θ(n) : n ∈ Z}. Our result answers the question raised by P. Sarnak in [8]. We also consider the sets {μ̂θ(rn) : n ∈ Z} for r > 0 which correspond to a linear change of variable for the measure. We show that such a set is still countable for all r ∈ Q(θ) but uncountable (a non-empty interval) for Lebesgue-a.e. r > 0.
منابع مشابه
SPECTRA OF BERNOULLI CONVOLUTIONS AS MULTIPLIERS IN Lp ON THE CIRCLE
It is shown that the closure of the set of Fourier coefficients of the Bernoulli convolution μθ parameterized by a Pisot number θ is countable. Combined with results of Salem and Sarnak, this proves that for every fixed θ > 1 the spectrum of the convolution operator f 7→ μθ∗f in L(S) (where S is the circle group) is countable and is the same for all p ∈ (1,∞), namely, {μ̂θ(n) : n ∈ Z}. Our resul...
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