Monday, January 1, 2014

The Fourier Series and Parsevals Theorem

1The Fourier series¶

• Let $f(x)$ be a periodic function on $2\pi$
• The series $s(x)$ is said to be the "Fourier series" associated to $f(x)$
• $s(x) = \frac{a_0}{2}+\sum_{n=1}^{\infty}(a_n \cos{nx} + b_n \sin{nx})$ where the Fourier coefficients are $a_n = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x) \cos{nx} ~dx$ and $b_n = \frac{1}{\pi}\int_{-\pi}^{\pi} f(x)\sin{nx} ~dx$ and $a_n, b_n$ are uniquely determined by $f(x)$

1.1Some properties¶

• $s(x) = f(x)$ whenever $f(x)$ is continuous at $x$
• $s(x) = \frac{f(x^+) + f(x^-)}{2}$ whenever $f(x)$ is discontinuous at $x$
• if $f(x)$ is odd then $a_n = 0 \forall n$
• if $f(x)$ is even then $b_n = 0 \forall n$

2Piece wise continuity¶

Let $f(x)$ be defined on $[a,b]$. We say that $f(x)$ is piece wise continuous on $[a,b]$ if:
$f(x)$ is continuous on $[a,b]$ except at finitely many points $x_1, x_2,\cdots$
At the discontinuity $x_1, x_2, \cdots x_n$ both one-sided limits exist

3Dirichlet Conditions¶

It is said that $f(x)$ will satisfy the Dirichlet conditions if:

• $f(x)$ is of period $2\pi$ and both $f$ and
• $f'$ are "piece wise continuous" on $[-\pi, \pi]$

4Parsevals Theorem¶

• If $f(x)$ satisfies the "Dirichlet conditions" then $\int_{-\pi}^{\pi} f^2(x) ~dx = \frac{(a_0)^2}{2} + \sum_{n=1}^{\infty} ((a_n)^2 + (b_n)^2)$