The norm of the basis functions are all $\sqrt{\pi}$, e.g.
$$
{\lVert cos(kx)\rVert} = \sqrt{\int_0^{2\pi}{cos^2(kx)dx}}
$$
$$
= \sqrt{\int_0^{2\pi}{{1+cos 2kx}\over2}}
$$
$$
= \sqrt{\pi}
$$
Therefore, $c_1$ is identical to the value in the slide
$$
c_1 = \int_0^{2\pi} {f(x) \cdot {cos(x) \over {\lVert cos(x)\rVert}}} dx
$$
$$
= \int_0^{2\pi} {1\over{\sqrt{\pi}}} f(x) cos(x) dx
$$
(Without normalization, we don't know where the $1\over{\sqrt{\pi}}$ in $c_1$ comes from.)
404Error_nf
The previous slide talks about sinusoidal orthonormal bases functions. Are there any other family of functions which form orthonormal bases? Also given a function, how do we decide what functions form the orthonormal bases set?
keenan
Are there any other family of functions which form orthonormal bases?
Sure; for instance, there are families of orthogonal polynomials. There are also periodic bases other than the sinusoids (though I already raised this question as a challenge on a different slide!)
There seems to be a flaw in the $c_{k}$ and $s_{k}$, as the basis functions haven't been normalized. After normalization, $c_k$ and $s_k$ are
$$ c_{k} = {\langle\langle f(x), {cos(kx) \over {\lVert cos(kx)\rVert}} \rangle\rangle} $$ $$ s_{k} = {\langle\langle f(x) , {sin(kx) \over {\lVert sin(kx)\rVert}} \rangle\rangle} $$
The norm of the basis functions are all $\sqrt{\pi}$, e.g. $$ {\lVert cos(kx)\rVert} = \sqrt{\int_0^{2\pi}{cos^2(kx)dx}} $$ $$ = \sqrt{\int_0^{2\pi}{{1+cos 2kx}\over2}} $$ $$ = \sqrt{\pi} $$ Therefore, $c_1$ is identical to the value in the slide $$ c_1 = \int_0^{2\pi} {f(x) \cdot {cos(x) \over {\lVert cos(x)\rVert}}} dx $$ $$ = \int_0^{2\pi} {1\over{\sqrt{\pi}}} f(x) cos(x) dx $$
(Without normalization, we don't know where the $1\over{\sqrt{\pi}}$ in $c_1$ comes from.)
The previous slide talks about sinusoidal orthonormal bases functions. Are there any other family of functions which form orthonormal bases? Also given a function, how do we decide what functions form the orthonormal bases set?
Sure; for instance, there are families of orthogonal polynomials. There are also periodic bases other than the sinusoids (though I already raised this question as a challenge on a different slide!)