P_r is defined as:

$\displaystyle P_r(x)=\frac{1-r^2}{1-2r\cos x +r^2}$

and

$\displaystyle P_r(x)=\frac{1-r^2}{1-2r\cos x +r^2}=\sum_{n=-\infty}^{\infty}r{|n|}e^{inx}$

and

$\displaystyle f(x)=\sum_{-\infty}^{\infty}c_ne^{inx}$

which is continues

i need to prove that:

$\displaystyle f_r(x)=\frac{1}{2\pi}\int_{-\pi}^{\pi}p_r(t)dt=\sum_{n=1}^{\infty}c_nr^{|n|}e^ {inx}$

the solution says to use the convolution property

$\displaystyle c_n(f)=c_n$

$\displaystyle c_n(P_r)=r^{|n|}$

$\displaystyle c_n(f_r)=c_n r^{|n|}$

but i cant see how the multiplication of those coefficient gives me the

expression i needed to prove

?

i only got the right side not the left integral

??