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For a continuous function $f$ of period $1,$ how to prove that $$\lambda_{n}=\frac{1}{n}\sum_{k=0}^{n-1}S_k(f),~~~~n \geq 1,$$ converges to $f$ pointwise, where $S_{k},~k \geq 0$ are partial sums of the Fourier series of $f.$

Starting with the Fourier series itslef, $f=\sum_{n \in \mathbb{Z}} c_n e^{i n t},$ where $c_n=\frac{1}{2 \pi} \int_{-\infty}^{\infty} f(t) \cdot e^{-int}~dt,$ and the partial sum is deifined accordingly. Does this have to do with a double sum and where do I need to use the periodicity, $f(t+1)=f(t)$ for all $t$ ? Is there a way to do this problem without using the Dirichlet kernel ? Any help in solving this problem is appreciated.

  • @i707107 This IS true, the question asks about Caesaro sums, not convergence of partial sums. – nullUser Sep 21 '16 at 03:47
  • Ah, that's right. Thanks for pointing it out. I was thinking about just pointwise convergence. – Sungjin Kim Sep 21 '16 at 03:48
  • No worries, I mistook that at first sight too. (Though the definition of $c_n$ is wrong and technically that needs to be fixed before it becomes true.) – nullUser Sep 21 '16 at 03:57

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