[Paper Review] Approximation of subharmonic functions

Igor Chyzhykov|ArXiv.org|Jul 13, 2008

Approximation Theory and Sequence Spaces3 citations

TL;DR

This paper establishes sharp L1-approximation estimates for subharmonic functions on the complex plane by logarithms of entire functions, under lower bounds on the Riesz measure. It proves that when the Riesz measure grows slowly but steadily (e.g., satisfying a condition involving a slowly varying function ψ), the approximation error in the L1 norm over disks of radius R is O(R² log ψ(R)), and this bound is optimal, generalizing earlier results by Lyubarskii and Malinnikova.

ABSTRACT

In certain classes of subharmonic functions u on C distinguished in terms of lower bounds for the Riesz measure of u, a sharp estimate is obtained for the rate of approximation by functions of the form log |f(z)|, where f is an entire function. The results complement and generalize those recently obtained by Yu. Lyubarskii and Eu. Malinnikova.

Motivation & Objective

• To improve upon existing L1-approximation estimates for subharmonic functions by entire functions' logarithmic moduli.
• To address the gap in approximation rates when the Riesz measure does not satisfy the strong lower bound condition of Lyubarskii and Malinnikova.
• To establish sharpness of the approximation error bound under natural growth conditions on the Riesz measure.
• To resolve M. Sodin’s question on whether a constant α ∈ [0,1) can improve the approximation rate in L1 norm.

Proposed method

• Introduces a class of slowly varying functions ψ to parameterize the growth of the Riesz measure in annuli.
• Uses integral estimates of the difference |u(z) − log|f(z)|| over disks |z| < R, integrating with respect to Lebesgue measure.
• Applies the Nevanlinna characteristic and counting functions to compare the zero distribution of f with the Riesz measure of u.
• Employs contradiction arguments via integral estimates of the form ∫ |n(t,f) − n(t,u) + α| / t dt to rule out existence of such f for α < 1/2.
• Constructs a model subharmonic function uψ(z) = ½∑ log|1 − z/rk| with Riesz measure satisfying the growth condition μ({R < |z| ≤ Rψ(R)}) > 1.
• Uses asymptotic analysis of the counting function and the Nevanlinna proximity function to derive lower bounds on the approximation error.

Experimental results

Research questions

• RQ1 Can the approximation error in L1 norm be improved when the Riesz measure satisfies a weak lower bound condition rather than a strong one?
• RQ2 Is the O(R² log ψ(R)) bound for the L1 approximation error sharp under the condition μ({R < |z| ≤ Rψ(R)}) > 1?
• RQ3 Can the approximation error be bounded uniformly in L1 norm with a logarithmic correction term log ψ(R) when the Riesz measure grows slowly?
• RQ4 Does the existence of an entire function f such that ∫|u − log|f|| dm = O(R²) hold under the given Riesz measure condition?
• RQ5 Is it possible to achieve a better approximation rate than O(R² log R) when the Riesz measure grows slower than any power?

Key findings

• For subharmonic functions u with Riesz measure satisfying μ({R < |z| ≤ Rψ(R)}) > 1 for a slowly varying function ψ, the L1 approximation error over |z| < R is O(R² log ψ(R)).
• The bound O(R² log ψ(R)) is sharp: no better rate is possible, as shown by constructing a model subharmonic function uψ satisfying the same measure condition.
• The approximation error cannot be bounded by O(R²) unless the Riesz measure grows at least as fast as a constant in annuli, which fails under the given ψ-growth condition.
• For α ∈ [0,1), no entire function f exists such that ∫|u − log|f| − α log|z|| dm = O(R²), even when the Riesz measure condition holds.
• The result generalizes and sharpens earlier results by Lyubarskii and Malinnikova, extending the approximation framework to subharmonic functions of arbitrary order.