While self studying Apostol Mathematical Analysis Chapter - Fourier Series , I have 2 questions in understanding a theorem whose image I am adding. (Question lines are highlighted)
Question 1: How can I prove that if $\sigma$ is a finite sum of trigonometric functions, it must generate a power series expansion which must converge uniformly on every finite interval.
Question 2 : I am not able to understand how defining p by p(x) = $ p_{ m} [ π ( x-a) / ( b-a) ]$ changes (31) to (30) which means I couldn't understand/ follow the last 2 lines .
Kindly guide.
As already mentioned in the comments, $\sin$ and $\cos$ can be expanded into power series of infinity radius of convergence. Surely the finite sum and composition of analytic functions, are analytic functions. Thus $\sigma$ is analytic.
Now for question number 2, $f(x)=g(\pi \frac{x-\alpha}{b-\alpha})$, for $x\in [\alpha , b]$. This is easy to verify by putting $t=\pi \frac{x-\alpha}{b-\alpha}$ and use the definition of $g$ (as proposed in the last line). Thus by $(31)$, by basically changing variables as it is suggested in the last line, we get $(30)$.