Let function $f(t)$ is represented by Fourier series,
$$\frac{a_0}{2}+\sum_{n=1}^{\infty}(a_n\cos{\frac{2n\pi t}{b-a}}+b_n\sin{\frac{2n\pi t}{b-a}}),$$
where $a$ and $b$ are lower and upper boundary,
$$a_0=\frac{2}{b-a}\int_{a}^{b}f(t)dt,$$
$$a_n=\frac{2}{b-a}\int_{a}^{b}f(t)cos\frac{2n\pi t}{b-a}dt,$$
$$b_n=\frac{2}{b-a}\int_{a}^{b}f(t)sin\frac{2n\pi t}{b-a}dt.$$
My question is, what conditions must be met so I can find derivative as (term by term)
$$\frac{d}{dt}\sum_{n=1}^{\infty}(a_n\cos{\frac{2n\pi t}{b-a}}+b_n\sin{\frac{2n\pi t}{b-a}})=\frac{d}{dt}(\sum_{n=1}^{\infty}a_n\cos{\frac{2n\pi t}{b-a}}+\sum_{n=1}^{\infty}b_n\sin{\frac{2n\pi t}{b-a}})=\sum_{n=1}^{\infty}\frac{d}{dt}(a_n\cos{\frac{2n\pi t}{b-a}})+\sum_{n=1}^{\infty}\frac{d}{dt}(b_n\sin{\frac{2n\pi t}{b-a}})?$$
Best Answer
A sufficient condition for differentiability of the series and commuting of sum and derivative is that $$ \sum_{n=1}^\infty n\big(\lvert a_n\rvert+\lvert b_n\rvert\big)<\infty. $$ See: Rudin, Principles of Mathematical Analysis, Theorem 7.17, p. 152.