Say you want to design a LP FIR filter with low pass cutoff $fc$, transition band $fc$ to $fs$ and ripple factor $dp$ at passband and $ds$ at stop band. If one divides the frequencies by $\pi$, then $0,fc,fs,\pi$ all get mapped from $[0,1]$. Since FIR filter is a polynomial, we can think of the filter as a polynomial map from $[0,1] \rightarrow [0,1]$. Kaiser's formula gives FIR filter length or the degree of the polynomial in terms of $fs-fc, ds$ and $dp$.
Now consider you are looking for a polynomial map from $[0,1]^n\rightarrow[0,1]$ so that a fixed subset of $[0,1]^n$ is the pass band and the complementary subset is the stop band. Is there a way to generalize Kasier's formula in this settings so that we can get an estimate on the total degree of the multivariate polynomial in this setting? Also what is design of such $n$-dimesnional filters (that is find the actual polynomials)?
