---
tag: 76JD0ZG3B1
kind: definition
name: Rational approximation
msc: [41A20, 41A21, 41A50]
defines: EHFQK1WXKS
uses: [0WCZH4AFMT, 7N4988HN3P, 0X3WGX95FE]
sources:
  - url: https://en.wikipedia.org/wiki/Rational_function
---

## Definition

Let $a < b$ be real numbers and let $m, n \ge 0$ be integers.
Let $R_{m,n}$ be the set of [[rational functions|0WCZH4AFMT]] $r = p/q$ with real polynomials $p$ and $q$ such that $\deg p \le m$ and $\deg q \le n$.
In addition, $q$ has no zero on $[a, b]$.
Let $f \colon [a, b] \to \R$ be a [[continuous function|0X3WGX95FE]].
A *rational approximation* of type $(m, n)$ to $f$ is an element $r$ of $R_{m,n}$ used in place of $f$.
Its error is $\norm{f - r}_\infty = \max_{x \in [a, b]} \abs{f(x) - r(x)}$.
The *error of best rational approximation* is
$$E_{m,n}(f) = \inf_{r \in R_{m,n}} \norm{f - r}_\infty.$$

## Remarks

For $n = 0$ this is polynomial approximation, and $E_{m,0}(f)$ is the error of [[best approximation|7N4988HN3P]] by polynomials of degree at most $m$.
For continuous $f$ on $[a, b]$, a best rational approximation in $R_{m,n}$ exists and is unique, and Chebyshev's equioscillation theorem characterizes it.
$R_{m,n}$ is not a vector space for $n \ge 1$, since a sum of two of its elements may need numerator degree $m + n$ and denominator degree $2n$.
This is why the theory differs from the linear case.
Example: $r(x) = 1/(1 + x^2)$ lies in $R_{0,2}$ on any interval, and $E_{0,2}(r) = 0$.
Non-example: on $[-1, 1]$, $1/x$ is not an element of $R_{0,1}$, since its denominator vanishes at $0 \in [-1, 1]$.