[Paper Review] Rudin's Theorem and Projective Hulls

John Wermer|ArXiv.org|Nov 2, 2006

Analytic and geometric function theory2 references3 citations

TL;DR

This paper establishes a sufficient condition under which a real-analytic function on the unit circle, when combined with holomorphic functions, generates a module in which all functions must be holomorphic. Using duality, the F. and M. Riesz theorem, and properties of analytic continuation, it proves that if point evaluations in the interior are bounded in the sup-norm over the boundary, then the function is holomorphic. The key result is that such a function must be entire if its graph's projective hull contains no extra points beyond the disk image.

ABSTRACT

Walter Rudin, in 1966, characterized analytic functions in the unit disk in terms of the maximum principle relative to the boundary. We conjecture a generalization of this result when the disk is replaced by the punctured disk, and we prove a special case of this conjecture, making use of the notion of projective hull in P^n introduced by R. Harvey and B. Lawson.

Motivation & Objective

• To extend Rudin’s Theorem from algebras to modules by introducing a bounded point evaluation condition.
• To investigate the structure of the projective hull of a closed curve in C^2 defined by a real-analytic function on the unit circle.
• To determine when the projective hull of such a curve equals the image of the unit disk under the function.
• To prove that if the projective hull contains no extra points beyond the disk image, then the function must be holomorphic (and entire).
• To establish a characterization of entire holomorphic functions via the projective hull of their graphs.

Proposed method

• Define a module M over the disk algebra A₀ as the set of functions a + bφ with a, b ∈ A₀ and φ continuous on Δ.
• Impose a bounded point evaluation condition: for each x ∈ int(Δ), |f(x)| ≤ Mₓ sup_Γ |f| for all f ∈ M.
• Use the Hahn-Banach theorem and the F. and M. Riesz theorem to represent point evaluation at 0 as integration against a measure σ on Γ.
• Show that the difference between σ and normalized Lebesgue measure is the boundary function of an H¹ function h.
• Apply the same technique to φσ and derive a representation φ = (α + k)/(1 + h) a.e. on Γ, where k and h are H¹ functions.
• Use analytic continuation and the fact that φ is real-analytic on Γ to show that (α + k)/(1 + h) extends to a meromorphic function on Δ, and then to an analytic function after multiplying by a finite Blaschke product.

Experimental results

Research questions

• RQ1 Under what conditions does a module over the disk algebra consisting of functions a + bφ with φ continuous on Δ and real-analytic on the boundary consist only of holomorphic functions?
• RQ2 Can the projective hull of a real-analytic curve γ in C² defined by (ζ, φ(ζ)) for ζ ∈ Γ contain points outside the image of the unit disk if φ is not holomorphic?
• RQ3 Is it true that if the projective hull of γ equals the image of the closed unit disk, then φ must be holomorphic on Δ?
• RQ4 What is the relationship between the boundedness of point evaluation functionals on M and the holomorphicity of φ?
• RQ5 Does the projective hull of the graph of a real-analytic function on the circle coincide with the image of the disk if and only if the function is entire?

Key findings

• If the evaluation functional at each interior point of Δ is bounded in the sup-norm over Γ, then every function in the module M is holomorphic.
• The function φ is shown to be equal a.e. on Γ to a ratio (α + k)/(1 + h), where k and h are H¹ functions, implying φ is meromorphic in Δ and thus holomorphic after removing poles.
• By analytic continuation and the real-analyticity of φ on Γ, the function φ extends to an entire holomorphic function on C.
• The projective hull of the curve γ = {(ζ, φ(ζ)) : ζ ∈ Γ} in C² equals the image of the closed unit disk if and only if φ is entire.
• A contradiction is derived by assuming φ is not holomorphic: a sequence of polynomials P_d of degree 2d is constructed such that |P_d| is small on γ but large at an interior point (α₀, φ(α₀)), violating the projective hull condition.