The idea of the proof of a theorem.

Asked Mar 09 '17 at 04:37

Active Mar 09 '17 at 04:37

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Could anyone explain the contradiction that he will use to prove the theorem?

Especially when he said that:" the idea now is to construct a family of trigonometric functions ${p_{k}}$ that peaks at 0, and so that $$\int p_{k}( \theta) f(\theta) d\theta \rightarrow \infty as\ k\ \rightarrow \infty.$$ This will be our contradiction, since those integrals tends to zero by assumption."

I do not see what is the assumption that makes them tends to zero and why the integral above tends to $\infty.$

Thanks. they are attached in the following photos.

asked Mar 09 '17 at 04:37

Emptymind

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The integrals $\int p_k(\theta) f(\theta) \ d\theta$ are all equal to zero (not merely "tend to zero"), because each $p_k$ is a trigonometric polynomial, and the Fourier coefficients of $f$ are zero by hypothesis. – Mar 09 '17 at 04:39
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This might be useful! – Error 404 Mar 09 '17 at 04:48
@VikrantDesai .... thank u sooo much :)) – Emptymind Mar 09 '17 at 05:03
Marking your question as a duplicate okay. ;) So that the answer will be found under a single hood. :) – Error 404 Mar 09 '17 at 05:07
@Bungo Thank u sooo much. – Emptymind Mar 09 '17 at 07:13
@Bungo ..... and why $$\int p_{k}( \theta) f(\theta) d\theta \rightarrow \infty as\ k\ \rightarrow \infty.$$ – Emptymind Mar 09 '17 at 07:43
@Bungo I mean why when { $p_{k}$} peaks at 0, when the previous integral tend to $\infty$ as k tend to $\infty$ – Emptymind Mar 09 '17 at 08:23

The idea of the proof of a theorem.

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