Orthogonal Projections

[Camillo]

a) Calculate in $L^2[0,1] $ the orthogonal projection of : $f(t) = sin(pit/2) $ on the subspace generated by $ x_1(t) = 1 ; x_2(t) = t $.

b) Let's be $V = L^2[-pi, pi] ; f(x) = | x| $; calculate the orthogonal projection $s_1(x) $ of $ f(x) $ on the subspace $F_1 $ generated by the functions : $ 1/2, cosx, sin x $ .
The function $s_1(x) $ is the best approximation of $ f(x) $ with elements of $ F_1 $.
Calculate, according to $ L^2 $ norm, the square of the distance between $f(x) $ and the subspace $F_1 $ .

[Camillo]

a) Correct .
It is non necessary to orthonormalize the basis $x_1,x_2 $ .
Another way is the following:
let's call $w =ax_1+bx_2$ the orthogonal projection of $f $ on the subspace; then $( f-w) $ is orthogonal to the subspace and consequently to each vector of the subspace , including $ x_1 $ as well as $x_2 $.
The conditions of orthogonality mean :
$<(f-w),x_1 > = 0 $ ; $<.,.> $ is the scalar product .
$<(f-w),x_2 > = 0 $
The solution of this system provides the required values of $a, b $.

b)Correct

[Kroldar]

"Camillo":
b) Let's be $V = L^2[-pi, pi] ; f(x) = | x| $; calculate the orthogonal projection $s_1(x) $ of $ f(x) $ on the subspace $F_1 $ generated by the functions : $ 1/2, cosx, sin x $ .
The function $s_1(x) $ is the best approximation of $ f(x) $ with elements of $ F_1 $.
Calculate, according to $ L^2 $ norm, the square of the distance between $f(x) $ and the subspace $F_1 $ .

We have to calculate the first $3$ terms of the Fourier series of the periodical replication of the signal $|x|$, $x in (-pi,pi)$. That's quite simple, so $s_1(x) = pi/2 - 4/pi cos(x)$.
The square of the distance between $f(x)$ and $F_1$ is given by
$int_(-pi)^(pi) [|x|-pi/2+4/pi cos(x)]^2 dx = (pi^4-96)/(6pi)$

[Kroldar]

"Camillo":a) Calculate in $L^2[0,1] $ the orthogonal projection of : $f(t) = sin(pit/2) $ on the subspace generated by $ x_1(t) = 1 ; x_2(t) = t $.

The signals $x_1(t) = 1$ and $x_2(t) = t$ are not orthogonal. By Gram-Schmidt's method, we can simply find two orthonormal signals which generate the same subspace generated by $x_1(t)$ and $x_2(t)$. These signals are $hat x_1(t) = 1$ and $hat x_2(t) = sqrt(12)(t-1/2)$.
The orthogonal projection of $f(t)$ on $hat x_1(t)$ is given by $int_0^1 sin(pi/2 t) dt = 2/pi$
The orthogonal projection of $f(t)$ on $hat x_2(t)$ is given by $int_0^1 sin(pi/2 t)sqrt(12)(t-1/2) dt = (2sqrt(3)(4-pi))/pi^2$
So the orthogonal projection of $f(t)$ on the subspace generate by $x_1(t)$ and $x_2(t)$ is
$2/pi + (2sqrt(3)(4-pi))/pi^2 sqrt(12)(t-1/2) = 2/pi + (6(4-pi)(2t-1))/pi^2$