Is it true that the sequences $ (A_{n})_{n \in \Bbb{N}} = (0)_{n \in \Bbb{N}} $ and $ (B_{n})_{n \in \Bbb{N}} = \left( \dfrac{1}{\sqrt{n}} \right)_{n \in \Bbb{N}} $ are the Fourier coefficients of some function $ f \in {L^{2}}[- \pi,\pi] $ with respect to the basis $ \left\{ \dfrac{1}{\sqrt{2}},\cos(nt),\sin(nt) \right\} $?
I’m a little confused about what this problem wants me to prove/disprove. How can I find the functions?
On
If such a function exists then $$\tag{1} f(t)=\frac1\pi\sum_{n=1}^\infty B_n\sin(nt). $$ But then we would have: \begin{eqnarray} \infty>\|f\|^2&=&\langle f,f\rangle=\frac{1}{\pi^2}\sum_{m,n}B_mB_n\int_{-\pi}^\pi\sin(mt)\sin(nt)\,dt=\frac{1}{\pi^2}\sum_{m,n}\pi B_mB_n\delta_{m,n}\\ &=&\frac1\pi\sum_{n=1}^\infty B_n^2=\frac1\pi\sum_{n=1}^\infty\frac1n=\infty, \end{eqnarray} which is a contradiction. Thus, such a function does not exist, in other word the Fourier series (1) does not converge to an element of $L^2(-\pi,\pi)$
In the Fourier transform represented by $C_n$ terms used by engineers this would more or less be equal to $i d(t)$ where $i$ is an imaginary unit and $d$ is a delta function. I conclude that the function in the coordinate domain was a flat offset with a phase factor.