# show $||f_x||_{L_p}=||f||_{L_p}$ for all p

let f be defined on [0,2$\pi$)

denote $f_x(y)=f(y-x)$

how do i go about showing $||f_x||_{L_p}=||f||_{L_p}$ for all p

i was trying to write out f in fourier series, and use the fact that the fourier coefficient of $f_x$ is $e^{inx}$ times the fourier coefficient of f. but i couldnt get anywhere.

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If I understand the question correctly, then $x \in [0,2 \pi]$ a fixed point. If so then it should be just a substitution in the integral defining the $L_p$ norm.