MATH 414 Lecture 13

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Convergence in the Mean

(also called $L^{2}$ convergence)

Orthogonal projections

Set-up:

1. Inner Product Space $V$, $\left\langle \cdot ,\cdot \right\rangle$
2. Subspace with orthonormal basis $V_{N}=\mathrm {Span} \left\{{\hat {u}}_{1},{\hat {u}}_{2},\ldots ,{\hat {u}}_{N}\right\}$

The orthogonal projection ${\vec {p}}$ of ${\vec {v}}\in V$ onto $V_{N}$ is the unique vector in $V_{N}$ such that $\left\|{\vec {v}}-{\vec {p}}\right\|=\min _{{\vec {w}}\in V_{N}}\left\|{\vec {v}}-{\vec {w}}\right\|$.

Properties:

1. ${\vec {p}}$ satisfies $\left\langle {\vec {v}}-{\vec {p}},{\vec {w}}\right\rangle =0$ for all ${\vec {w}}\in V_{N}$.
2. ${\vec {p}}=\sum _{j=1}^{N}\left\langle {\vec {v}},{\hat {u}}_{j}\right\rangle \,{\hat {u}}_{j}$
3. $\left\|{\vec {v}}-{\vec {p}}\right\|^{2}=\left\|{\vec {v}}\right\|^{2}-\left\|{\vec {p}}\right\|^{2}$

Application to Fourier Series

With Fourier Series, we project a function $f$ onto the vector space $V_{N}=\mathrm {Span} \left\{{\frac {1}{\sqrt {2\pi }}},{\frac {1}{\sqrt {\pi }}}\,\cos {t},{\frac {1}{\sqrt {\pi }}}\,\sin {t},\ldots ,{\frac {1}{\sqrt {\pi }}}\,\cos {(N\,t)},{\frac {1}{\sqrt {\pi }}}\,\sin {(N\,t)}\right\}$ (this basis is indeed orthonormal.

The $L^{2}$ error of this projection is $\left\|f-S_{n}\right\|^{2}=\left\|f\right\|^{2}-2\pi \,a_{0}^{2}-\pi \left(\sum _{k=1}^{N}a_{k}^{2}+b_{k}^{2}\right)$

Theorem. $S_{n}$ converges to $f$ in the mean if and only if

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

Proof. $\int _{-\pi }^{\pi }\left|f-S_{N}\right|^{2}\,\mathrm {d} t=E_{N}^{2}=\int _{-\pi }^{\pi }\left|f\right|^{2}\,\mathrm {d} t-2\pi \left\{{\frac {a_{0}^{2}}{2}}+\sum _{k=1}^{N}\left(a_{k}^{2}+b_{k}^{2}\right)\right\}$. If $E_{N}^{2}$ converges to 0, then $\sum _{k=1}^{N}\left(a_{k}^{2}+b_{k}^{2}\right)$ must also converge to 0.

quod erat demonstrandum

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Analysis and Synthesis

Given $f(t)$, it's easy to come up with a Fourier series (Analysis of signal)

However, Given $a_{0}$, $a_{n}$, and $b_{n}$, is there a $f(t)$ such that $f(t)=a_{0}+\sum _{n=1}^{\infty }a_{n}\,\cos {(n\,t)}+b_{n}\,\sin {(n\,t)}$? (Synthesize to get a signal)

Answer. Yes! As long as ${\frac {a_{0}}{2}}+\sum _{n=1}^{\infty }a_{n}^{2}+b_{n}^{2}$ is finite, but this requires $L^{2}$ integrals (Riesz-Fisher Theorem) because Riemann integrals don't work.