Chapter 6: Rational Approximation Theory

By definition, a rational function is the quotient of two polynomials; and by Theorem 4.3.4$\text{<a href="../pdf/notes.pdf\#loc-244">Theorem 4.3.4</a>}\text{<a href="../weierstrass-theory/entireness-and-meromorphy/index.html\#thm-theorem-4-3-4">Theorem 4.3.4</a>}$, in equivalent formulation, it is a function meromorphic on all of $\widehat{ℂ}$. The poles and zeros may not accumulate in $\widehat{ℂ}$, and thus there are finitely many as a consequence of Bolzano–Weierstrass (Theorem 1.1.10$\text{<a href="../pdf/notes.pdf\#loc-12">Theorem 1.1.10</a>}\text{<a href="../prerequisites-calculus-topology/topology/index.html\#thm-theorem-1-1-10">Theorem 1.1.10</a>}$).

When we refer to approximation, we refer to the approximation of a function as the uniform limit (of a sequence) of functions. Let $𝐾\subseteq ℂ$ be compact and suppose $𝑓:𝐾\to ℂ$ is a given function on $𝐾$. As a consequence of Mergelyan’s Theorem (Theorem 6.2.7$\text{<a href="../pdf/notes.pdf\#loc-415">Theorem 6.2.7</a>}\text{<a href="mergelyans-theorem/index.html\#thm-theorem-6-2-7">Theorem 6.2.7</a>}$), sufficient conditions for $𝑓$ to be the uniform limit of rational functions whose poles lie in (a subset of given points of) $\widehat{ℂ}\setminus 𝐾$ are the continuity of $𝑓$ on $𝐾$ and the holomorphy of $𝑓$ on $\stackrel{⚬}{𝐾}$.