User:EGM6341.S11.team5.cavalcanti/Mtg5

Mtg 5: Fri, 14 Jan 11 [[media: Nm1.s11.mtg5.djvu | Page 5-1]]

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Scalar (inner product) of 2 functions (cont'd)

Consider

$$\sum _{i}\underbrace{f\left({x}_{i}\right)g\left({x}_{i}\right)}_{\color{blue}{u(x_{i})}}\underbrace{h}_{\color{blue}{h \to 0 as n \to \infty }}\color{red}{< +\infty} $$

“finite” for integrable functions

Another similar choice:

$$< f,g> \mathrm{\colon }=\frac{h}{2}u\left({x}_{0}\right)+\sum _{i=\color{red}{1}}^{\color{red}{n-1}}u\left({x}_{i}\right)h+\frac{h}{2}u\left({x}_{n}\right)\color{blue}{={T}_{n}\left(u\right)_{n \to \infty}}$$

$$\color{blue}{{T}_{n}\left(u\right)}$$ = Trapezoidal rule for $$\color{blue}{{\int }_{a}^{b}\underbrace{u\left(x\right)}_{f(x)g(x)}\mathit{dx}}$$

[[media: Nm1.s11.mtg5.djvu | Page 5-2]]  As n → ∞,

2 – Norm of f:

→ Orthogonal functions →

e.g., $$\color{blue}{\left\lbrace \mathrm{1,}\cos n\omega x,\sin n\omega x,\mathrm{...}\right\rbrace }$$

in Fourier series expansion

Legendre polynomial

Chebyshev polynomial, etc....

Runge phenomenon

Gauss-Legendre quadrature

Spectral methods (Clenshaw-Curtis quadrature)

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Proof of Taylor Series: Similar technique used in error analysis later.