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Paul M. Gauthier

Publications and source records attributed to Paul M. Gauthier.

10 recordsLinked to original sources
arXiv2023

Radial limits of solutions to elliptic partial differential equations

For certain elliptic differential operators $L,$ we study the behaviour of solutions to $Lu=0,$ as we tend to the boundary along radii in strictly starlike domains in $\R^n, n\ge 3.$ Analogous results are obtained in other special domains. Our approach involves introducing harmonic line bundles as instances of Brelot harmonic spaces and approximating continuous functions by harmonic functions on appropriate subsets. These approximation theorems on harmonic spaces yield interesting examples for approximation by solutions of $Lu=0$ on some domains in $\R^n.$

math.AP↗
arXiv2021

Asymptotic first boundary value problem for holomorphic functions of several complex variables

In 1955, Lehto showed that, for every measurable function $ψ$ on the unit circle $\mathbb T,$ there is function $f$ holomorphic in the unit disc $\mathbb D,$ having $ψ$ as radial limit a.e. on $\mathbb T.$ We consider an analogous boundary value problem, where the unit disc is replaced by a Stein domain on a complex manifold and radial approach to a boundary point $p$ is replaced by (asymptotically) total approach to $p.$

math.CV↗
arXiv2021

Asymptotic first boundary value problem for elliptic operators

In 1955, Lehto showed that, for every measurable function $ψ$ on the unit circle $\mathbb T,$ there is a function $f$ holomorphic in the unit disc, having $ψ$ as radial limit a.e. on $\mathbb T.$ We consider an analogous problem for solutions $f$ of homogenous elliptic equations $Pf=0$ and, in particular, for holomorphic functions on Riemann surfaces and harmonic functions on Riemannian manifolds.

math.CV↗
arXiv2020

Approximation by random complex polynomials and random rational functions

We investigate random compact sets with random functions defined thereon, such as polynomials, rational functions, the pluricomplex Green function and the Siciak extremal function. One surprising consequence of our study is that randomness can be used to `improve' convergence for sequences of functions.

math.CV↗
arXiv2019

An algebra of polyanalytic functions

The most important uniform algebra is the family of continuous functions on a compact subset $K$ of the complex plane $\mathbb{C}$ which are analytic on the interior int$(K)$ For compact sets $K$ which are regular (i.e. $K =$int$(K)$ and for polyanalytic functions, we introduce analogous spaces, which are Banach spaces with respect to the sup-norm, but are not closed with respect to the usual pointwise multiplication. We shall introduce a multiplication on these spaces and investigate the resulting algebras.

math.CV↗