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Let $K_N$ the Fejer's kernel on $\mathbb{T}$. Let $l$ be a positive integer. Let $Q$ the function defined by $$ Q(t)=K_N(lt). $$ In Hewitt/Ross "Abstract Harmonic Analysis 2" page 438, I can read that if $1<p<2$ we have $$ ||Q||_{L_p}=||K_N||_{L_p}. $$ Why?

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In general, if you have a function $f$ on $\mathbb{T}$ and an integer $k$, $$ \int_{\mathbb{T}}f(t)\,\mathrm{d}t=\int_{\mathbb{T}}f(kt)\,\mathrm{d}t $$ The integral on the right is just $k$ copies of the integral on the left compressed $k$ times (so each copy has $1/k$ times the integral on the left).

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