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 Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x)} .

Proof. [to be discussed later]

quod erat demonstrandum

#18 Convolution

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (f * g)(x) := \frac{1}{2\pi} \, \int_{-\pi}^{\pi} f(t) \, g(x-t) \,\mathrm{d}t}

Suppose this has the fourier series Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sum_{n=-\infty}^\infty \sigma_n \, \mathrm{e}^{i \, n \, x}} , where Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \sigma_n = \frac{1}{2\pi} \, \int_{-\pi}^{\pi} (f*g)(x) \, \mathrm{e}^{-i \, n \, x} \,\mathrm{d}x}

BIG HINT: replace Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle (f*g)(x)} with its definition

Fubini's theorem will be useful when changing integrals.

Parseval's Theorem

Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f \in L^2 \left[ -\pi, \pi \right]} . This implies that Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f} has finite energy over a finite interval.

An Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle L^2} equality:

• Let Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = a_0 + \sum_{n=1}^\infty a-n \, \cos{(n\,x)} + b_n \, \sin{(n\,x)}}
• Alternatively, Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f(x) = \sum_{n=-\infty}^\infty c_n \, \mathrm{e}^{i \, n \, x}}

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 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:

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