Proof of Fejer's lemma (exercise from book)

Question

Does someone know where I can find the proof of the following lemma? The exercise is from Katznelson's Introduction To Harmonic Analysis. Thanks in advance for help.

Answer 1

Following the hint: for $\varepsilon >0$ there exists a trigonometric polynomial $P(x)=\sum_{k=-N}^Nc_ke^{-ikx}$ such that $\|P-f\|_1<\varepsilon.$ Then $$\left |{1\over 2\pi}\int\limits_{0}^{2\pi}f(t)g(nt)\,dt -{1\over 2\pi}\int\limits_{0}^{2\pi }P(t)g(nt)\,dt\right |\\\le \|g\|_\infty{1\over 2\pi}\int\limits_{0}^{2\pi} |f(t)-P(t)|\,dt<\|g\|_\infty\,\varepsilon$$ Moreover $$|\hat{f}(0)\hat{g}(0)-\hat{P}(0)\hat{g}(0)|=|\hat{f}(0)-\hat{P}(0)|\,|\hat{g}(0)|<\|g\|_\infty\,\varepsilon$$ By the triangle inequality we get $$\left |{1\over 2\pi}\int\limits_{0}^{2\pi }f(t)g(nt)\,dt -\hat{f}(0)\hat{g}(0)\right |\\ \le \left |{1\over 2\pi}\int\limits_{0}^{2\pi }P(t)g(nt)\,dt -\hat{P}(0)\hat{g}(0)\right |+2\|g\|_\infty\,\varepsilon$$ For $k\neq 0$ we have $${1\over 2\pi}\int\limits_{0}^{2\pi} e^{-ikt}g(nt)\,dt=\begin{cases}\hat{g}(k/n)& k/n\in\mathbb{Z}\\ 0 & {\rm otherwise}\end{cases}$$ Thus $${1\over 2\pi}\int\limits_{0}^{2\pi} e^{-ikt}g(nt)\,dt=0,\quad n>|k| $$ For $k=0$ we get $${1\over 2\pi}\int\limits_{0}^{2\pi }g(nt)\,dt={1\over 2\pi}\int\limits_{0}^{2\pi }g(t)\,dt=\hat{g}(0)$$ Hence $${1\over 2\pi}\int\limits_{0}^{2\pi }P(t)g(nt)\,dt=\hat{P}(0)\hat{g}(0),\quad n>N $$ Summarizing for $n>N$ we obtain $$\left |{1\over 2\pi}\int\limits_{0}^{2\pi }f(t)g(nt)\,dt -\hat{f}(0)\hat{g}(0)\right |\le 2\|g\|_\infty\,\varepsilon$$