2 Questions in proof of theorem 11.17 Apostol ( Mathematical Analysis)

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.

• Both $\sin, \cos$ have power series expansions that are converge uniformly on any bounded interval. If two power series converge uniformly on any bounded interval then so does their sum and hence any finite sum. Jun 25, 2020 at 22:13
• @copper.hat how to prove that power series expansion of sin x converge uniformly on any bounded interval? Can you give a formal proof? Also can you give any explaination for part (b) ?
– user775699
Jun 26, 2020 at 6:47

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 $$(30)$$.