Theoretical Understanding of Fourier Series

Question

I understand mathematically how it is derived, and how it works (ie how it is applied and the significance of Fourier Series in solving PDEs). What I want to know is why it works, theoretically and mathematically. I am unable to prove to myself exactly why a Fourier Series converges on an interval as opposed to a Taylor Series, which is only at a point. Can someone explain theoretically why a Fourier Series does what it does (or via proof)?

Answer 1

I don't know the exact reason about Fourier Series coefficients should only be found piecewisely by finite intervals, but I know the principle of finding formulae of Fourier Series coefficients are only as simple as follows:

For $f(x)=\int_a^bF(u)K(u,x)~du$ or $f(x)=\sum\limits_{u=a}^bF(u)K(u,x)$ ,

Consider $f(x)K(v,x)=\int_a^bF(u)K(u,x)K(v,x)~du$ or $f(x)K(v,x)=\sum\limits_{u=a}^bF(u)K(u,x)K(v,x)$

$\int_p^qf(x)K(v,x)~dx=\int_p^q\int_a^bF(u)K(u,x)K(v,x)~du~dx$ or $\int_p^qf(x)K(v,x)~dx=\int_p^q\sum\limits_{u=a}^bF(u)K(u,x)K(v,x)~dx$ or $\sum\limits_{x=p}^qf(x)K(v,x)=\sum\limits_{x=p}^q\int_a^bF(u)K(u,x)K(v,x)~du$ or $\sum\limits_{x=p}^qf(x)K(v,x)=\sum\limits_{x=p}^q\sum\limits_{u=a}^bF(u)K(u,x)K(v,x)$

$\int_p^qf(x)K(v,x)~dx=\int_a^bF(u)\int_p^qK(u,x)K(v,x)~dx~du$ or $\int_p^qf(x)K(v,x)~dx=\sum\limits_{u=a}^bF(u)\int_p^qK(u,x)K(v,x)~dx$ or $\sum\limits_{x=p}^qf(x)K(v,x)=\int_a^bF(u)\sum\limits_{x=p}^qK(u,x)K(v,x)~du$ or $\sum\limits_{x=p}^qf(x)K(v,x)=\sum\limits_{u=a}^bF(u)\sum\limits_{x=p}^qK(u,x)K(v,x)$

If we can find $p$ and $q$ so that either $\int_p^qK(u,x)K(v,x)~dx=C$ or $\sum\limits_{x=p}^qK(u,x)K(v,x)=C$ of which $C\neq0$ and independent of $u$ and $v$ when $u=v$ under the integral range of $a$ and $b$ or the summation range of $a$ and $b$ ,

Then we can say that either $F(v)C=\int_p^qf(x)K(v,x)~dx$ or $F(v)C=\sum\limits_{x=p}^qf(x)K(v,x)$ holds for at least $x\in(p,q)$

i.e. either $F(v)=\dfrac{1}{C}\int_p^qf(x)K(v,x)~dx$ or $F(v)=\dfrac{1}{C}\sum\limits_{x=p}^qf(x)K(v,x)$ holds for at least $x\in(p,q)$

i.e. either $F(u)=\dfrac{1}{C}\int_p^qf(x)K(u,x)~dx$ or $F(u)=\dfrac{1}{C}\sum\limits_{x=p}^qf(x)K(u,x)$ holds for at least $x\in(p,q)$