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The Hardy-Littlewood maximal function ℳ and the trigonometric function sin x are two central objects in harmonic analysis. We prove that ℳ characterizes sin x in the following way: Let be a periodic function and α > 1/2. If there exists a real number 0 < γ < ∞ such that the averaging operator
has a critical point at r = γ for every x ∈ ℝ, then
f(x) = a + bsin(cx+d) for some a,b,c,d ∈ ℝ.
This statement can be used to derive a characterization of trigonometric functions as those nonconstant...
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