MATH 414 Lecture 12

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Three Types of Periodic Extensions

Standard $\pi$-periodic (a regular Fourier Series)

Even $2\pi$-periodic extension (Cosine Series)

Odd $2\pi$-periodic extension (Sine Series)

Uniform Convergence

Let $f$ be a $2\pi$-periodic function with Fourier Series $a_{0}+\sum _{n=1}^{\infty }a_{n}\,\cos {(n\,x)}+b_{n}\,\sin {(n\,x)}$. If $S_{N}(x)=a_{0}+\sum _{n=1}^{N}a_{n}\,\cos {(n\,x)}+b_{n}\,\sin {(n\,x)}$, then $S_{N}$ converges uniformly to $f$ on $x\in \mathbb {R}$ if and only if for every $\epsilon >0$, there is a $N_{0}$ that does not depend on $x$ such that $\left|f(x)-S_{N}(x)\right|<\epsilon$ for all $x$, provided $N>N_{0}$.

Theorem. If $f$ is piecewise smooth and continuous (no jumps, but it can have corners, then $S_{N}(x)$ converges uniformly to $f(x)$.

Proof. [to be discussed later]

quod erat demonstrandum

#18 Convolution

$(f*g)(x):={\frac {1}{2\pi }}\,\int _{-\pi }^{\pi }f(t)\,g(x-t)\,\mathrm {d} t$

Suppose this has the fourier series $\sum _{n=-\infty }^{\infty }\sigma _{n}\,\mathrm {e} ^{i\,n\,x}$, where $\sigma _{n}={\frac {1}{2\pi }}\,\int _{-\pi }^{\pi }(f*g)(x)\,\mathrm {e} ^{-i\,n\,x}\,\mathrm {d} x$

BIG HINT: replace $(f*g)(x)$ with its definition

Fubini's theorem will be useful when changing integrals.

Parseval's Theorem

Let $f\in L^{2}\left[-\pi ,\pi \right]$. This implies that $f$ has finite energy over a finite interval.

An $L^{2}$ equality:

• Let $f(x)=a_{0}+\sum _{n=1}^{\infty }a-n\,\cos {(n\,x)}+b_{n}\,\sin {(n\,x)}$
• Alternatively, $f(x)=\sum _{n=-\infty }^{\infty }c_{n}\,\mathrm {e} ^{i\,n\,x}$

$\int _{-\pi }^{\pi }\left|f(t)\right|^{2}\,\mathrm {d} t=2\pi \,\sum _{n=-\infty }^{\infty }\left|c_{n}\right|^{2}=2\pi \,\left|a_{0}\right|^{2}+\pi \,\sum _{n=1}^{\infty }\left|a_{n}\right|^{2}+\left|b_{n}\right|^{2}$

Example

$f(x)=\pi -x$. Consider the even extension of $f$ from $-\pi$ to $\pi$.

$\pi -x={\frac {\pi }{2}}+4\,\sum _{k=1}^{\infty }{\frac {\cos {((2k-1)\,x)}}{(2k-1)^{2}}}$

1. $\int _{-\pi }^{\pi }\left|f(x)\right|^{2}\,\mathrm {d} x=2\int _{0}^{\pi }\left(\pi -x\right)^{2}\,\mathrm {d} x={\frac {2}{3}}\,\pi ^{3}$
2. $a_{0}={\frac {\pi }{2}}$, $b_{n}=0$, $a_{odd\ n}=0$

$\int _{-\pi }^{\pi }\left|f(x)\right|^{2}\,\mathrm {d} x={\frac {2}{3}}\,\pi ^{3}=2\pi \,\left({\frac {\pi }{2}}\right)^{2}+\left(\sum _{k=1}^{\infty }{\frac {16}{(2k-1)4}}\right)\,\pi$

Consider the odd extension of $f$ from $-\pi$ to $\pi$.

$\pi -x=\sum _{n=1}^{\infty }{\frac {2}{n}}\sin {(n\,x)}$

1. $a_{n}=0$, $b_{n}={\frac {2}{n}}$

$\int _{-\pi }^{\pi }\left|f(x)\right|^{2}\,\mathrm {d} x=2\,\int _{0}^{\pi }\left(\pi -x\right)^{2}\,\mathrm {d} x={\frac {2}{3}}\,\pi ^{3}=\pi \,\left(\sum _{n=1}^{\infty }{\frac {4}{n^{2}}}\right)$

We get ${\frac {\pi ^{2}}{6}}=\sum _{n=1}^{\infty }{\frac {1}{n^{2}}}$

Punchline

If $f(t)=\sum c_{n}\,\mathrm {e} ^{i\,n\,t}$, then the total energy in the wave is equal to the sum of energies of all its modes:

${\frac {1}{2\pi }}\,\int _{-\pi }^{\pi }\left|f(t)\right|^{2}\,\mathrm {d} t=\sum _{n=-\infty }^{\infty }\left|c_{n}\right|^{2}$