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제 14강. 함수공간

Function Spaces and Fourier Series

무한대 공간으로 생각하는 것이다. 셀 수 없이 많은 공간을 $H$ , 힐버트 스페이스라고 말한다.

Vector inner product -> function inner product
$x^{T} y = \sum_{i = 1}^{n} x_{i} y_{i}$ -> $(x (t), y (t)) = \sum_{k = 0}^{\infty} x (a + k δ t) y (a + k δ t) = \int_{a}^{b} x (t) y (t) d t$

크기(거리)도 똑같다.

$| | x (t) | |^{2} = \int_{a}^{b} x^{2} (t) d t$ (적분화)

Hilbert space is a vector space in $R^{\infty}$ where vectors have finite lengths

For continuous functions, the summation for length $| | f | |$ is replaced with integration in a interval

$f (x) = x \in R^{\infty}$

$| | f (x) | |^{2} = \int_{0}^{2 n} x \times x d x = π < \infty$

$X^{T} y = 0$ = 수직이다

$(x (t), y (t)) = \int_{a}^{b} x (t) y (t) d t$ = 수직이다

$X = \sum_{i = 1}^{n} a_{i} v_{i}$ -> $x (t) = \sum_{i = 1}^{\infty} a_{i} b_{i} (t)$

$a_{i}$ 의 계수 = $\sum (q_{i}^{T} a) q_{i}$ -> $\sum_{i = 1}^{\infty} (q_{i} (t) x (t)) q_{i} (t)$

Orthogonal basis functions ; { $\sin n x, \cos m x$ } n,m : integer

$f (x) = a_{0} + \sum_{n = 1}^{\infty} a_{n} \cos n x + \sum_{m = 1}^{\infty} b_{m} \sin m x$

$a_{1} = \frac{\int f (x) \cos x d x}{(\cos x \cos x)}$

=> Fourier Series is a Projection onto $\cos n x$ and $\sin m x$

Polynomial functions : $1, x, x^{2} \dots$ are independent ,but not orthogonal for $0 \leq x \leq 1$

change interval -> -1 ≤ x ≤ 1

$a_{j} - \sum_{i = 1}^{j - 1} (q_{i}^{T} a_{j}) q_{i} ⊥ q_{i} (i = 1, 2, 3, \dots, j - 1)$

(1) $b_{1} (x) = 1$

(2) $x - (q_{1} (x) x) \times 1$ = $x - \int_{- 1}^{1} x d x \times 1$ = $q_{2} (x)$

$(b_{1} (x), b_{2} (x)) = (1, x) = \int_{- 1}^{1} x d x = 0$ -> Orthogonal

(3) $b_{3} (x) = x^{2} - (q_{1} (x) x^{2}) q_{1} (x) - (q_{2} (x) x^{2}) q_{2} (x)$

= $x^{2} - \int_{- 1}^{1} x^{2} d x \times 1 - \int_{- 1}^{1} x^{2} x d x \times x$

= $x^{2} - \frac{1}{3}$

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