[Paper Review] Two-radii theorem for solutions of some mean value equations

Olga D. Trofimenko|arXiv (Cornell University)|Sep 16, 2013

Functional Equations Stability Results4 references3 citations

TL;DR

This paper establishes a two-radii theorem for solutions of integral mean value equations involving spherical means in complex analysis. It characterizes solutions of the poly-analytic equation $(\partial_z)^{m-s}(\partial_{\bar{z}})^m f = 0$ via a representation formula in terms of Bessel functions and entire functions, proving that such solutions are uniquely determined by their mean values over two distinct radii when the corresponding zero sets of auxiliary functions are disjoint.

ABSTRACT

A description of solutions of some integral equations has been obtained. A two-radii theorem is obtained as well.

Motivation & Objective

• To characterize smooth solutions of a class of integral mean value equations involving spherical means in the complex plane.
• To establish a two-radii theorem that links solutions of a poly-analytic differential equation to their integral mean values over two fixed radii.
• To determine conditions under which a function satisfying mean value conditions for two radii must be a solution of a specific poly-analytic PDE.
• To analyze the structure of solutions via Fourier-Bessel series and entire functions with controlled growth.

Proposed method

• Derive a representation formula for solutions of the mean value equation (1) in terms of Fourier coefficients and special functions $ \Phi_{\lambda,\eta,k}(\rho) $, which are derivatives of Bessel functions.
• Define the function $ g_r(z) $, whose zeros determine the spectral components of the solution, and introduce the set $ Z_r $ of zeros in the right half-plane with non-negative imaginary part.
• Use Paley-Wiener theory and convolution equations with compactly supported distributions $ T_1, T_2 $ associated with $ g_{r_1}, g_{r_2} $ to analyze the solution space.
• Apply theorems on convolution equations and zero sets of entire functions to relate the vanishing of convolutions to the structure of solutions.
• Use the condition $ Z(r_1, r_2) = \emptyset $ to deduce that the only solutions are those satisfying the poly-analytic PDE (2).
• Construct counterexamples when $ Z(r_1, r_2) \neq \emptyset $, showing that mean value conditions for two radii do not imply the PDE unless the zero sets are disjoint.

Experimental results

Research questions

• RQ1 Under what conditions does a function satisfying a mean value equation for two distinct radii necessarily satisfy a poly-analytic PDE?
• RQ2 How can the solution space of a mean value equation be fully characterized in terms of Bessel functions and entire functions?
• RQ3 What is the role of the zero set $ Z(r_1, r_2) $ in determining whether two-radii mean value conditions imply a differential equation?
• RQ4 Can a smooth function satisfy mean value conditions for two radii without being a solution of the corresponding poly-analytic PDE?
• RQ5 What is the precise structure of solutions to the integral mean value equation (1) in terms of Fourier-Bessel series and special functions?

Key findings

• The solution space of the mean value equation (1) for a fixed radius $ r $ is completely described by a finite linear combination of monomial terms and an infinite series involving $ \Phi_{\lambda,\eta,k}(\rho) $, with coefficients decaying rapidly as $ |\lambda| \to \infty $.
• When $ Z(r_1, r_2) = \emptyset $, any function satisfying the mean value condition for both $ r_1 $ and $ r_2 $ must be smooth and satisfy the poly-analytic PDE $ (\partial_z)^{m-s}(\partial_{\bar{z}})^m f = 0 $.
• If $ Z(r_1, r_2) \neq \emptyset $, there exist non-trivial smooth functions satisfying the two-radii mean value condition that do not solve the PDE, showing the necessity of the zero set condition.
• The function $ \Phi_{\lambda,\eta,k}(\rho) $, defined as a derivative of $ J_k(z\rho) $, forms a complete basis for the solution space when combined with polynomial terms.
• The asymptotic behavior of zeros of $ g_r(z) $, including logarithmic bounds on imaginary parts and lower bounds on the derivative, ensures the convergence and uniqueness of the solution representation.
• The proof relies on distributional convolution and Paley-Wiener theory, showing that the vanishing of $ f * T_1 $ and $ f * T_2 $ in overlapping domains implies $ f $ is a solution of the PDE if and only if $ Z(r_1, r_2) = \emptyset $.