Let $$a_k = \frac{1}{k\pi}\sum_{j=0}^m\alpha_j\sin(kT_j)$$ $\alpha_j,T_j \in \mathbb{R}, 0 I'd like to know the limit if it exists, $$\lim_{n\to\infty}\frac{1}{\log(n)}\sum_{k=1}^n\left|a_k\right|$$
A question on a limit arising in Fourier series
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real-analysis
limits
summation
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0If $x_n = \frac{\sum_{k=1}^n |a_k|}{\log(n)}$ then using $|a_n| \leq \frac{C}{n}$ we get that $x_n$ is bounded. Unfortunately $x_n$ is not monotone in general. It's tempting to try Stolz–Cesàro however this leads to a limit which does not exist in general so not very helpful. Probabilistic methods might be useful. If $m=0$ and $X = (kT_0 \mod 2\pi)/2\pi$ is treated as a uniform random variable on $[0,1]$ then since $E[|\sin(2\pi X)|] = \frac{2}{\pi}$ we should have something like $\frac{2|\alpha_0|}{\pi^2}$ if the limit exist (and $T_0$ is not a rational multiple of $\pi$). – 2017-01-30
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0@Winther Can this be useful, creates some hope: $$\int_0^{\lambda}\left|\frac{\sin t}t\right|dt \sim C\log(\lambda)$$ – 2017-01-31
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0Can we say : $$\int_0^{\lambda}\left|\frac{A\sin(at) + B\sin(bt) + C\sin(ct)}t\right|dt \sim K\log(\lambda)$$ – 2017-01-31