I believe I have a naive or hard question because I couldn't find any results in the Internet yet. Any help is greatly appreciated.

So suppose I have two rational functions $R_1(x)$ and $R_2(x)$, i.e., $R_i(x) = P_i(x,m_i)/Q_i(x,n_i)$ where

$$P_i(x,m_i) = p_{i,0} + p_{i,1}x + \dots + p_{i,m_i} x^{m_i}$$ $$Q_i(x,n_i) = q_{i,0} + q_{i,1}x + \dots + q_{i,n_i} x^{n_i}$$

are polynomials in $x$ with degree $m_i$ (or $n_i$ respectively). Next, suppose that $R_1(x)$ is *fixed* and I want to approximate $R_1(x)$ with $R_2(x)$ where the polynomials in $R_2(x)$ have strictly lower degree, i.e., $m_2 < m_1$ and $n_2 < n_1$. I also know that the polynomials $Q_i(x)$ in the denominator have strictly larger degree that the numerator polynomials $P_i(x)$. In fact, $n_i \ge m_i+2$. Furthermore, by "approximation" I mean that I want to minimize the integral

$${\cal A} = \min_{\textbf{p}_2,\textbf{q}_2}\int_a^b dx [R_1(x)-R_2(x)]^2,$$

where $\textbf{p}_2 = (p_{2,0},\dots,p_{2,m_2})$ and $\textbf{q}_2 = (q_{2,0},\dots,q_{2,n_2})$ is the set of free parameters in $R_2(x)$ and $[a,b]$ is some interval (I am happy for an answer for any interval). Now my two questions are:

1) What is the best approximation? Is it simply to choose $\textbf{p}_2 = (p_{1,0},\dots,p_{1,m_2})$ and $\textbf{q}_2 = (q_{1,0},\dots,q_{1,n_2})$, i.e., the same coefficients as in $R_1(x)$? That would be the simple answer.

2) If there is no simple answer, are there at least bounds known for the *minimum* error in the approximation? That is to say, I would like to know lower bounds of the form ${\cal A} \ge \dots$, but in the literature usually the opposite is discussed (i.e., upper bounds on the error ${\cal A} \le \dots$).

Technical sideremark: I know that the rational functions $R_1(x)$ is sufficiently "well-behaved" in the sense that its Fourier transform exists and also all coefficients of $R_1(x)$ can be assumed to be real.

Thanks a lot. Any idea or hint is greatly appreciated.

Kind regards, Philipp