I've come across the Fejer kernel $K_N(t) = \sum_{k = -N}^N (1- \frac{\vert k \vert}{N+1}) e^{ikt}$. Now, in my application, I only need the positive part, that is $K_n^{+}(t) =\sum_{k=0}^N (1- \frac{\vert k \vert}{N +1}) e^{ikt}$. Now, for the Fejer kernel itself one can show using by the representation $$ K_n(t) = \frac{1}{n+1} \big (\frac{\sin(\frac{n+1}{2}t)}{\sin (\frac{1}{2}t)} \big )^2 $$ that $$ \lim_{n \rightarrow \infty} \frac{1}{2\pi} \int_{\delta}^{2\pi - \delta} \vert K_n(t) \vert dt = 0 $$ holds for all $ 0 < \delta < \pi$. What I'm actually interested in is whether "something similar" is possible for $K_n^+$. Since this is rather vague I've just asked for a closed form in the title of this question because I think a closed form would be enough for me, and I could think about the rest by myself.


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