MATH 414 Lecture 12

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Three Types of Periodic Extensions

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

Even 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 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

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 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} from 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 -\pi} 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 \pi} .

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

1. 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(x) \right|^2 \,\mathrm{d}x = 2 \int_0^\pi \left( \pi-x \right)^2 \,\mathrm{d}x = \frac{2}{3} \, \pi^3}
2. 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 a_0 = \frac{\pi}{2}} , 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 b_n = 0} , 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 a_{odd\ n} = 0}

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(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 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} from 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 -\pi} 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 \pi} .

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

1. 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 a_n = 0} , 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 b_n = \frac{2}{n}}

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(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 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 \frac{\pi^2}{6} = \sum_{n=1}^\infty \frac{1}{n^2}}

Punchline

If 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(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:

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 \frac{1}{2\pi} \, \int_{-\pi}^\pi \left| f(t) \right|^2 \,\mathrm{d}t = \sum_{n=-\infty}^\infty \left| c_n \right|^2}