Mergelyan's theorem

mergelyan_theorem

Submitter: Kim Morrison.

Notes: Mergelyan's theorem: if K ⊆ ℂ is compact with connected complement, then every function continuous on K and holomorphic on interior K is uniformly approximable on K by complex polynomials. §64 of Knill's 'Some Fundamental Theorems in Mathematics'.

Source: S. N. Mergelyan, *On the representation of functions by series of polynomials on closed sets*, Doklady Akad. Nauk SSSR 78 (1951), 405-408 (Russian). Listed as §64 (additional statement 4) in O. Knill, *Some Fundamental Theorems in Mathematics* (https://people.math.harvard.edu/~knill/graphgeometry/papers/fundamental.pdf).

Informal solution: A standard proof approximates f uniformly on K by an entire function, using Cauchy–Green / Pompeiu or related ∂̄ methods, and then approximates the entire function uniformly on the compact set K by Taylor polynomials.

theorem declaration uses `sorry`mergelyan (K : Set ℂ) (_hK : IsCompact K) (_hKc : IsConnected (Kᶜ))
    (f : ℂ → ℂ) (_hfc : ContinuousOn f K) (_hfh : AnalyticOnNhd ℂ f (interior K))
    (ε : ℝ) (_hε : 0 < ε) :
    ∃ p : ℂ[X], ∀ z ∈ K, ‖f z - p.eval z‖ < ε := byK:Set ℂ_hK:IsCompact K_hKc:IsConnected Kᶜf:ℂ → ℂ_hfc:ContinuousOn f K_hfh:AnalyticOnNhd ℂ f (interior K)ε:ℝ_hε:0 < ε⊢ ∃ p, ∀ z ∈ K, ‖f z - Polynomial.eval z p‖ < ε
  sorryAll goals completed! 🐙

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