PX153 - J2 - convergence

f_{N} (x) = \frac{a_{0}}{2} + \sum_{n = 1}^{\infty} (a_{n} \cos (n x) + b_{n} \sin (n x))

for $- π \leq x \leq π$

minimum requirement (convergence in the mean)

any function for which $\int_{- π}^{π} (f (x))^{2} d x < \infty$ can be expressed as a fourier series
a weak form of convergence:

lim_{N \to \infty} f_{N} (x) \neq f (x)

at all $- π \leq x \leq π$

if $f (x)$ and $f^{'} (x)$ are continuous at $[- π, π)$ , except possibly at a finite number of points, the fourier series converges at every point, $\tilde{f} (x) = \frac{1}{2} (f (x^{+}) + f (x^{-}))$ , where, $f (x^{+}) = lim_{ϵ \to 0} f (x + ϵ)$ , $f (x^{-}) = lim_{ϵ \to 0} f (x - ϵ)$ , and $ϵ$ is a scalar
eg:

lim_{N \to \infty} f_{N} (x) = \tilde{f} (x) = \frac{1}{2} (f (x^{+}) + f (x^{-}))

- at discontinuities, the series converges to the midpoints

what happens at the boundaries? ie: at $x = π$ and $- π$
- note:
$f_{N} (x + 2 π) = \frac{a_{0}}{2} + \sum_{n = 1}^{\infty} (a_{n} \cos (n x + 2 π) + b_{n} \sin (n x + 2 π))$ $f_{N} (π) = f_{N} (- π + 2 π) = f_{N} (- π)$

\begin{aligned} lim_{N \to \infty} f_{N} (- π) & = lim_{ϵ \to 0} \frac{1}{2} [f (- π + ϵ) + f (- π - ϵ)] \\ = lim_{ϵ \to 0} \frac{1}{2} [f (- π + ϵ) + f (π - ϵ)] \end{aligned}

- so, the series converges at the average of the original function at the boundaries

if $f (x)$ and $f^{'} (x)$ exist and are continuous everywhere in $[- π, π)$ , and $f (π) = f (- π)$ , the fourier series converges uniformly to $f (x)$
$| f (x) - f_{N} (x) |$ has a maximum value within $[- π, π)$ for given $N$ , which is useful if $1 %$ precision is needed, and $lim_{N \to \infty} | f (x) - f_{N} (x) | = 0$
why is this needed?
- sines and cosines are well behaved (differentiate and integrate infinite number of times), and are easy to work with
- natural separation between slowly varying terms with $x$ (small $n$ ), and rapidly varying terms with $x$ (large $n$ )
- many systems in physics involve waves. eg: light, particles in QM, electronics, signal propagation, etc