I'm having trouble understanding a couple of parts of the proof presented here in Knapp's Basic Real Analysis : Book link from Google (pgs 389-390).

He defines . He then takes the Fourier series for L=1 on and gets which seemingly identifies . This means to me that is an even function of t, but this is not mentioned; am I missing something here?

My second question is when he seemingly substitutes in the integral However shouldn't the exponential in the integrand be ?

Thank you for any help!