Precisely how does the performance of an FIR filter depend on its design criteria?
To illustrate, here is a specific question that probably does not have a convenient answer:
The Parks-McClellan algorithm converges to a FIR filter optimally aligned to design parameters. With it one can design a low-pass filter with minimum pass- and stop-band ripple:
Take $\mathcal{S}_n$ to be the collection of symmetric FIR filters of order $n$: \begin{align} \mathcal{S}_n := \left\{(h_1,\dots,h_n)\in \mathbb{R}^n: h_k=h_{n-k+1} \text{ for each }k\right\}. \end{align} Write the frequency response of $h\in \mathcal{S}_n$ as $\hat{h}(\omega)=\sum_k h_ke^{-j2\pi \omega k}.$
Specify a "stopband weight" $\alpha > 0$, cutoff $\omega>0$ and transition width $\varepsilon>0$. Define filter mismatch as follows: \begin{align} \gamma_n(h; \alpha,\omega,\epsilon)\! := \min_{f\in [0,\omega]} \left|1-|\hat{h}(f)|\right| + \alpha\cdot \max_{f\in[\omega+\epsilon, 2\pi]} |\hat{h}(f)|. \end{align} The Parks-McClellan algorithm with parameters $(\alpha, \omega,\varepsilon)$ converges to an $h_{best}$ which minimizes $\gamma_n(h; \alpha, \omega, \varepsilon)$ over all $\mathcal{S}^n$.
What estimating expressions depending on $(n, \alpha,\omega, \varepsilon)$ provide tight upper and lower bounds for $\min_h \gamma_n(h;\alpha, \omega,\varepsilon)$ ?
Alternatively, fixing $n,\ (\alpha,\omega, \varepsilon)$ and $\Delta > 0$, are there bounds for $\min_h \max_{f\in[\omega+\epsilon, 2\pi]} |\hat{h}(f)|$ among those $h\in \mathcal{S}_n$ that have $\min_{f\in [0,\omega]} \left|1-|\hat{h}(f)|\right| \leq \delta$
