[Paper Review] A Remark on Two Extensions of the Disc Algebra and Mergelyan's Theorem

Vassili Nestoridis, Ioannis Papadoperakis|arXiv (Cornell University)|Apr 5, 2011

Meromorphic and Entire Functions1 references3 citations

TL;DR

This paper extends Mergelyan's Theorem to uniform approximation on closed Jordan domains using the chordal metric χ and an alternative compactification of the complex plane. It proves that the classes of functions $χ$-uniform limits of polynomials on such domains coincide with generalized disc algebra extensions, including functions with spherical or harmonic infinity-type boundary behavior.

ABSTRACT

We investigate the set of uniform limits of polynomials on any closed Jordan domain with respect to the chordal metric $χ$ on $\mathbb{C}\cup\{\infty \}$. We conclude that Mergelyan's Theorem may be extended to the case of uniform approximation with respect to $χ$ on closed Jordan domains. Similar results are obtained if we replace the one point compactification $\mathbb{C}\cup\{\infty\}$ of $\mathbb{C}$ by another compactification of $\mathbb{C}$ homeomorphic to the closed unit disc.

Motivation & Objective

• To extend Mergelyan’s classical theorem on polynomial approximation to the setting of uniform convergence with respect to the chordal metric χ on closed Jordan domains.
• To investigate whether the class of uniform χ-limits of polynomials on closed Jordan domains coincides with a generalized disc algebra that includes functions with infinite values.
• To analyze the case of an alternative compactification of ℂ, homeomorphic to the closed unit disc, where infinity is represented as a circle of points.
• To characterize the set of uniform limits of polynomials on closed Jordan domains under this new compactification and metric.
• To establish that both generalized function classes arise as uniform limits of polynomials under their respective metrics.

Proposed method

• Utilize the Riemann mapping theorem to transfer the problem from a general Jordan domain Ω to the unit disc D via a conformal map φ: D → Ω.
• Define the generalized class $χ$-A(Ω) as the set of functions f: $χ$-A(Ω) → ℂ ∪ {∞} that are continuous on $χ$-A(Ω), holomorphic in Ω, and have radial limits in ℂ ∪ {∞} at the boundary.
• Apply the known result that $χ$-A(D) equals the set of uniform χ-limits of polynomials on $χ$-A(D), using the chordal metric χ on ℂ ∪ {∞}.
• Use the triangle inequality and metric comparison (χ(a,b) ≤ |a−b|) to relate χ-convergence on Ω to uniform convergence on D and back to polynomial approximation.
• Introduce an alternative compactification of ℂ via the map z ↦ z/(1+|z|), inducing a metric d on ℂ ∪ {∞·e^{iθ}}.
• Define the class $χ$-A(Ω) as functions continuous on $χ$-A(Ω), holomorphic in Ω, and of infinite type with g(z) = ∞·e^{iθ(z)} where θ is continuous on $χ$-A(Ω) and harmonic in Ω.

Experimental results

Research questions

• RQ1 Can Mergelyan’s Theorem be extended to uniform approximation with respect to the chordal metric χ on closed Jordan domains?
• RQ2 Does the class of uniform χ-limits of polynomials on a closed Jordan domain Ω coincide with the generalized disc algebra $χ$-A(Ω)?
• RQ3 What is the structure of the set of uniform limits of polynomials on Ω under an alternative compactification of ℂ where ∞ is replaced by a circle of points?
• RQ4 How does the harmonic boundary behavior of the argument of infinity (θ(z)) affect the approximation properties in the extended function class?
• RQ5 Is the class of functions with harmonic infinity-type boundary values equal to the set of uniform limits of polynomials under the metric d?

Key findings

• The class $χ$-A(Ω) of functions continuous on $χ$-A(Ω), holomorphic in Ω, and with boundary values in ℂ ∪ {∞}, coincides with the set of uniform χ-limits of polynomials on $χ$-A(Ω).
• For any function g ∈ $χ$-A(Ω), there exists a sequence of polynomials Q_n such that sup_{z∈$χ$-A(Ω)} χ(Q_n(z), g(z)) → 0 as n → ∞.
• The proof relies on transferring the problem to the unit disc via a Riemann map and using the known χ-approximation property of polynomials in $χ$-A(D).
• The alternative compactification of ℂ induces a metric d under which the class $χ$-A(Ω) is also characterized as the uniform d-limits of polynomials on $χ$-A(Ω).
• Functions of infinite type in $χ$-A(Ω) are of the form g(z) = ∞·e^{iθ(z)} where θ is continuous on $χ$-A(Ω) and harmonic in Ω.
• The result establishes a direct analog of Mergelyan’s Theorem in the context of spherical and compactified complex approximation, extending classical polynomial approximation to include functions with infinite boundary values.