I'm trying to workout an example of a textbook on undergrad functional analysis I'm studying but something's just not working out. Probably missing on something really basic. I'm gonna state some results first:
Proposition: Let $\mathscr{S}=\{e_i:i\in \mathbb{N}\}$ be an orthonormal set in an inner product space $E$. If $u=\sum_{i=1}^\infty \xi _ie_i$, then $u_i=\langle u,e_i\rangle =\xi _i$ and $\|u\|^2=\sum_{i=1}^\infty |\xi _i|^2$, where $u_i$ is the Fourier coefficient of $u$ in the $e_i$ direction.
Theorem (Riesz-Fischer): Let $\mathscr{S}=\{e_i:i\in \mathbb{N}\}$ be an orthonormal set in a Hilbert space $\mathscr{H}$. Given a sequence $(\xi _i)\in \mathbb{K}$, the series $\sum_{i=1}^\infty \xi _ie_i$ converges if and only if the series $\sum_{i=1}^\infty |\xi _i|^2$ converges.
So now we get an orthonormal system$$\mathscr{S}=\left \{\frac{1}{\sqrt{2\pi}},\frac{\cos t}{\sqrt{\pi}}, \frac{\sin t}{\sqrt{\pi}},\frac{\cos 2t}{\sqrt{\pi}},\frac{\sin 2t}{\sqrt{\pi}},\ldots \right \}$$on the Hilbert space $L^2([-\pi ,\pi ],\mathbb{R})$. Choose constants $a_0,a_1,b_1,a_2,b_2,\ldots$ such that$$\frac{a_0^2}{2}+\sum_{k=1}^\infty \left (a_k^2+b_k^2\right )<\infty .$$Now, the textbook says that by the Riesz-Fischer Theorem there exists $f\in L^2([-\pi ,\pi ],\mathbb{R})$ given by$$f=a_0\frac{1}{\sqrt{2\pi}}+\sum_{k=1}^\infty \left (a_k\frac{\cos kt}{\sqrt{\pi}}+b_k\frac{\sin kt}{\sqrt{\pi}}\right ).$$I think this should be pretty obvious and direct but I'm getting a different answer. Since $\xi _0^2=|\xi _0|^2=\frac{a_0^2}{2}$, I'm getting this:$$f=a_0\frac{1}{2\sqrt{\pi}}+\sum_{k=1}^\infty \left (a_k\frac{\cos kt}{\sqrt{\pi}}+b_k \frac{\sin kt}{\sqrt{\pi}}\right ).$$Later, in another example, they refer to $\mathscr{S}$ saying that it forms an orthonormal basis for $L^2([-\pi ,\pi],\mathbb{R})$ and says that every function $f$ in this space can be written by$$f(t)=\frac{a_0}{2}+\sum_{n=1}^\infty a_n\cos nt+b_n\sin nt.$$What is happening? Shouldn't it be the same as earlier? Is the textbook wrong or am I going crazy?
In both examples they state that$$a_k=\frac{1}{\sqrt{\pi}}\int_{-\pi}^\pi f(t)\cos kt\,dt,\quad k\in \{0,1,2,\ldots \}$$and$$b_k=\frac{1}{\sqrt{\pi}}\int_{-\pi}^\pi f(t)\sin kt\,dt,\quad k\in \mathbb{N},$$which made sense to me.
I haven't found another reference that makes a similar treatment on the subject, if you guys have any recommendations it'd be great! Thanks in advance!
The constants are basically irrelevant. But the easiest convention is the one where the resulting basis is orthonormal, which is $$ \left\{ \frac{1}{\sqrt{2\pi}},\frac{\cos(x)}{\sqrt{\pi}},\frac{\sin(x)}{\sqrt{\pi}},\frac{\cos(2x)}{\sqrt{\pi}},\frac{\sin(2x)}{\sqrt{\pi}},\cdots\cdots \right\}. $$ This is the correct normalization for an orthonormal basis of $L^2[0,2\pi]$, which is undoubtedly why they are using their normalization. Then the ordinary Fourier series for $f\in L^2[0,2\pi]$ is an expansion along an orthonormal basis of $L^2[0,\pi]$: $$ f=\langle f,\frac{1}{\sqrt{2\pi}}\rangle\frac{1}{\sqrt{2\pi}}+\langle f,\frac{\cos(x)}{\sqrt{\pi}}\rangle\frac{\cos(x)}{\sqrt{\pi}}+\langle f,\frac{\sin(x)}{\sqrt{\pi}}\rangle\frac{\sin(x)}{\sqrt{\pi}}+\cdots \\ = \frac{1}{2\pi}\int_0^{2\pi}f(x')dx'+\frac{1}{\pi}\int_0^{2\pi}f(x')\cos(x')dx'\cos(x)+\frac{1}{\pi}\int_0^{2\pi}f(x')\sin(x')dx'\sin(x)+\cdots. $$