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This seems like something one might find in a book so I would be grateful for any references you think may be helpful.

I am interested in the rate at which of a function integrated against the $N$th Fejer Kernel approaches it's value at zero. In other words, I am looking to understand the big oh term below $$\int_{-\pi}^\pi f(t)F_N(t) dt - f(0) = O(1/N^?)$$ where $$F_N(t) = \frac{1}{2\pi}\cdot\frac{\sin^2(Nt/2)}{N\sin^2(t/2)} = \sum_{n = -N+1}^{N-1}\left(1-\frac{|n|}{N}\right)\cos(nt)$$ is the $N$th Fejer Kernel and a delta-sequence. I can't find this result anywhere. Thank you in advance.

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    $\begingroup$ Without any conditions on $f$, there may be no rate at all. What are your conditions on $f$? $\endgroup$
    – Iosif Pinelis
    Commented Jun 16, 2023 at 13:54
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    $\begingroup$ You can get some rate of convergence by assuming that f is Holder continuous. $\endgroup$
    – an_ordinary_mathematician
    Commented Jun 16, 2023 at 13:57
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    $\begingroup$ @IosifPinelis I since came across a related question which demonstrates what an_ordinary_mathematician said. I was mostly interested in smooth, periodic functions, which being Lipchitz, can be shown from the related question to have an error of at most big oh 1/N. I was hoping one could do better than this for smooth, periodic functions but as Carlo Beenakker demonstrated, in general, this is not the case. $\endgroup$
    – JMcB
    Commented Jun 16, 2023 at 17:47

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For any $\delta,x$ such that $0<\delta\leq |x|<\pi$ one has $|F_N(x)|\leq[2\pi(N+1)\sin^2(\delta/2)]^{-1}$, so the error in the delta-function approximation is of order $1/N$.

Two explicit examples, if $f(t)=\cos t$ one has $$\int_{-\pi}^\pi f(t)F_N(t) dt =1-\frac{1}{N}+{\cal O}(N^{-2}).$$ and if $f(t)=\sin^2(t/2)$ one has $$\int_{-\pi}^\pi f(t)F_N(t) dt =\frac{1}{2N}.$$

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    $\begingroup$ "so the error in the delta-function approximation is of order $1/N$." Why? First of all, you said nothing about the contribution of $x$ with $|x|<\delta$. Second, the rate (and even the very existence of any nontrivial rate) will of course depend on $f$. $\endgroup$
    – Iosif Pinelis
    Commented Jun 16, 2023 at 13:51
  • $\begingroup$ @Carlo Beenakker, thank you for pointing out these two examples! $\endgroup$
    – JMcB
    Commented Jun 16, 2023 at 17:51

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