Define $w: \Bbb R \rightarrow \Bbb C$ by
\begin{equation} w(t) =\begin{cases} 1/\sqrt{2\pi} & t \in [0, 2\pi)\\ 0 & \text{otherwise}. \end{cases} \end{equation}
and for $n \in \Bbb Z$ and $m \in \Bbb Z$, define $\varphi_{n,m} \in L^2(\Bbb R)$ by $\varphi_{n,m}(t) = w(t-2n\pi)e^{imt}$.
Show $\Phi = \{\varphi_{n,m} | n,m \in \Bbb Z\}$ is an orthonormal basis for $L^2(\Bbb R)$ using standard facts about Fourier Series.
Let $\Psi = \varphi_{n,m}(t) = w(t-n\pi)e^{imt}$. Show $\Psi$ is not a basis in $L^2(\Bbb R)$ by giving two distinct expansions for the zero signal term.
Find frame bounds for $\Psi$ and the dual frame for $\tilde{\Psi}$. Express an arbitrary signal $x \in L^2(\Bbb R)$ in terms of this frame and, in particular, express $w$ as an expansion in $\Psi$.
My Attempt/Input
The first two parts, I kind of get intuitively but am having a hard time formalizing it. I am a bit lost on the last part. I know what the condition of a frame is and that a tight frame is when the bounds A=B but don't know how to show that.