Pade approximation of a rational function

Question

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

Answer 1

Mean square approximation of a continuous function by rational functions on an interval is studied in the monograph

Walsh, J. L. Interpolation and approximation by rational functions in the complex domain. American Mathematical Society Colloquium Publications, Vol. XX American Mathematical Society, Providence, R.I., 1965

and the paper

Cheney, E. W.; Goldstein, A. A. Mean-square approximation by generalized rational functions. Math. Z. 95, 1967, 232-241.

You can find a proof of the existence of a best approximant in the above book by Walsh. Let $R_{m,n}$ denote the set of rational functions $p/q$ with $\deg p\leq m$ and $\deg q\leq n$. By differentiation, it can be established that a best approximant $r_{0}=p_{0}/q_{0}\in R_{m,n}$ to a function $f$ must satisfy the set of non-linear critical equations $$\int_{a}^{b}(f-r_{0})\frac{\mathbb{P}_{k}}{q_{0}^{2}}dt=0,$$ where $k=\max(m+\deg q,n+\deg p)$ and $\mathbb{P}_{k}$ denotes the space of polynomials of degree at most $k$. These critical equations may admit several different rational functions as solutions, among which local maxima, saddle points, and local minima. Regarding your questions,

1) the fact that the function you want to approximate is itself a rational function does not make the problem easier. Of course, there is no reason that you get a best approximant by ``truncating'' the coefficients. If the degrees involved in your problem are not too big, you may try to solve numerically the critical equations.

2) If you do not specify the function you want to approximate (outside of being rational), it is certainly impossible to estimate the error in approximation. For the particular case of the signum function on $[-1,1]$, upper and lower bounds have been obtained in

N.S. Vjacheslavov, On the least deviations of the function ${\rm sgn}~x$ and its primitives from the rational functions in the $L^{p}$ metrics, $0<p<\infty$, Mat. Sb. 103 (1977); Math. USSR Sb. 32 (1977), pp. 19-31.