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Search: MSC category 32A15 ( Entire functions )

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1. CMB Online first

Li, Bao Qin
An Equivalent Form of Picard's Theorem and Beyond
This paper gives an equivalent form of Picard's theorem via entire solutions of the functional equation $f^2+g^2=1$, and then its improvements and applications to certain nonlinear (ordinary and partial) differential equations.

Keywords:entire function, Picard's Theorem, functional equation, partial differential equation
Categories:30D20, 32A15, 35F20

2. CMB 2011 (vol 55 pp. 249)

Chang, Der-Chen; Li, Bao Qin
Description of Entire Solutions of Eiconal Type Equations
The paper describes entire solutions to the eiconal type non-linear partial differential equations, which include the eiconal equations $(X_1(u))^2+(X_2(u))^2=1$ as special cases, where $X_1=p_1{\partial}/{\partial z_1}+p_2{\partial}/{\partial z_2}$, $X_2=p_3{\partial}/{\partial z_1}+p_4{\partial}/{\partial z_2}$ are linearly independent operators with $p_j$ being arbitrary polynomials in $\mathbf{C}^2$.

Keywords:entire solution, eiconal equation, polynomial, transcendental function
Categories:32A15, 35F20

3. CMB 2009 (vol 53 pp. 11)

Burke, Maxim R.
Approximation and Interpolation by Entire Functions of Several Variables
Let $f\colon \mathbb R^n\to \mathbb R$ be $C^\infty$ and let $h\colon \mathbb R^n\to\mathbb R$ be positive and continuous. For any unbounded nondecreasing sequence $\{c_k\}$ of nonnegative real numbers and for any sequence without accumulation points $\{x_m\}$ in $\mathbb R^n$, there exists an entire function $g\colon\mathbb C^n\to\mathbb C$ taking real values on $\mathbb R^n$ such that \begin{align*} &|g^{(\alpha)}(x)-f^{(\alpha)}(x)|\lt h(x), \quad |x|\ge c_k, |\alpha|\le k, k=0,1,2,\dots, \\ &g^{(\alpha)}(x_m)=f^{(\alpha)}(x_m), \quad |x_m|\ge c_k, |\alpha|\le k, m,k=0,1,2,\dots. \end{align*} This is a version for functions of several variables of the case $n=1$ due to L. Hoischen.

Keywords:entire function, complex approximation, interpolation, several complex variables

4. CMB 1997 (vol 40 pp. 129)

Badea, Catalin
Sur les caractères d'une algèbre de Banach
A new proof for the Gleason-Kahane-\.Zelazko theorem concerning the characters of a Banach algebra is given. A theorem due to P\'olya and Saxer is used instead of the Hadamard factorization theorem.

Categories:46H05, 32A15

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