Mergelyan's theorem

Mergelyan's theorem states that every function continuous on a compact subset of the complex plane and holomorphic in its interior is uniformly approximable there by polynomials if the complement is connected.

Uniform convergence: A sequence of functions converges uniformly when one error bound works across the entire domain. The theorem guarantees approximation in this global sense, not merely pointwise closeness.
Runge's theorem: Runge's theorem characterizes rational approximation of holomorphic functions through the location of poles. It supplies approximation methods that can be adapted to establish Mergelyan’s polynomial conclusion.
Weierstrass approximation theorem: The Weierstrass approximation theorem says every continuous real-valued function on a closed interval is uniformly approximable by polynomials. Unlike Mergelyan’s theorem, it needs no holomorphicity because its domain is a real interval.
Polynomial approximation: Polynomial approximation represents functions by polynomials with error small in a specified sense. Mergelyan’s theorem is a central criterion for when this approximation works uniformly on compact planar sets.
Holomorphic function: A complex-valued function is holomorphic on an open set if it is complex differentiable at every point there. Holomorphicity in the compact set’s interior is the theorem’s analytic hypothesis.
Cauchy integral formula: The Cauchy integral formula expresses holomorphic function values through an integral over a surrounding contour. Integral representations help convert analytic information into approximating polynomials.
Stone–Weierstrass theorem: The Stone–Weierstrass theorem gives conditions under which an algebra of functions is dense among continuous functions. It offers an abstract density framework, while Mergelyan gives a sharp planar criterion for polynomials.
Numerical analysis: Numerical analysis studies algorithms for computing approximate solutions to mathematical problems. Polynomial approximants can provide computable surrogates for functions satisfying the theorem’s hypotheses.
Compact set: A compact set is one for which every open cover has a finite subcover. The approximation is uniform on this set, whose compactness controls the domain.
Mollifier: A mollifier is a smooth, localized averaging function used to regularize less regular functions. Smoothing can help separate boundary continuity from interior analytic approximation in proof strategies.

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