[[concept]]Theorem
Let be -bandlimited and a sequence converging to . Then the GFT converges to the WFT
Proof
Let be the set of eigenvalues in the -bandlimited set. Then for all , we have
\lvert (\hat{x}_{n})_{i} - \hat{x}_{i} \rvert &= \lvert \langle X_{n}, \varphi_{i}^{(n)} \rangle - \langle {\cal X}, \varphi_{i}\rangle \rvert \\ &= \lvert \langle X_{n}, \varphi_{i}^{(n)} \rangle + \langle {\cal X}, \varphi_{i}^{(n)}\rangle - \langle {\cal X}, \varphi_{i}^n\rangle - \langle {\cal X}, \varphi_{i}\rangle \rvert \\ &\leq \lvert \lvert {\cal X} - X_{n} \rvert \rvert \cdot \lvert \lvert \varphi_{i}^{(n)} \rvert \rvert + \lvert \lvert {\cal X} \rvert \rvert \cdot \lvert \lvert \varphi_{i}^{(n)} - \varphi_{i} \rvert \rvert \end{aligned}$$ By convergence of $X_{n}$, for all $\epsilon>0$, there exists some $n_{1}$ such that $\lvert \lvert X_{n} - {\cal X} \rvert \rvert \leq \frac{\epsilon}{2}$ for all $n > n_{1}$. And for $\lvert \lvert \varphi_{i}^{(n)} -\varphi_{i}\rvert \rvert$, we have $$\lvert \lvert \varphi_{i}^{(n)} -\varphi_{i}\rvert \rvert \leq \frac{\pi}{2} \frac{\lvert \lvert T_{W}-T_{W_{n}} \rvert \rvert }{d(\lambda_{i}, \lambda_{i}^{(n)})} \leq \frac{\pi}{2} \frac{\lvert \lvert T_{W} -T_{W_{n}} \rvert \rvert }{\min_{i \in {\cal C}} d(\lambda_{i}, \lambda_{i}^{(n)})}$$ And from $T_{W_{n}} \to T_{W}$, for all $\epsilon>0$, there is some $n_{2}$ such that $\lvert \lvert \varphi_{i}^{(n)} -\varphi_{i}\rvert \rvert \leq \frac{\epsilon}{2}$ for all $n>n_{2}$ Thus, for any $\epsilon$ and each $n > \max \{ n_{1}, n_{2} \}$ and for all $i \in {\cal C}$, we have $$\lvert (\hat{x}_{n})_{i} - \hat{x}_{i} \rvert \leq \lvert \lvert X_{n} - {\cal X} \rvert \rvert\cdot \cancelto{1}{\lvert \lvert \varphi_{i}^{(n)} \rvert \rvert} + \lvert \lvert {\cal X} \rvert \rvert \cdot \lvert \lvert \varphi_{i}^{(n)} - \varphi_{i} \rvert \rvert \leq \frac{\epsilon}{2} + \lvert \cancel{\lvert{\cal X} \rvert \rvert} \frac{\epsilon}{2 \cancel{\lvert \lvert {\cal X} \rvert \rvert} } = \epsilon $$ And for $i \not\in {\cal C}$, $$\langle \varphi_{i}(T_{W_{n}}), X_{n} \rangle \to \langle \Psi, {\cal X} \rangle = 0$$ Where $\Psi \in \perp \text{span}\{ \varphi_{i}: i \in {\cal C} \}$ (orthogonal complement) $\blacksquare$
Mentions
Mentions
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