arXiv subjects
Publications and source records attributed to Muhammed Ali Alan.
In this paper, we prove that a compact set $K\subset \mathbb{C}^n$ is the support of a weighted equilibrium measure if and only it is not pluripolar at each of its points extending a result of Saff and Totik to higher dimensions. Thus, we characterize the supports weighted equilibrium measures completely. Our proof is a new proof even in one dimension.
In this paper we give two complete characterizations of the Poletsky- Stessin- Hardy spaces in the complex plane: First in terms of their boundary values as a weighted subclass of the usual $L^p$ class with respect to the arclength measure on the boundary. Second we completely describe functions in these spaces by having a harmonic majorant with a certain growth condition and we prove some basic results about these spaces. In particular, we prove approximation results in such spaces and extend the classical result of Beurling which describes the invariant subspaces of the shift operator. Additionally we provide non-trivial examples.
We analyze the supports of weighted equilibrium measures in $\mathbb{C}^n$. We give explicit examples of families of compact sets which arise as the support of a weighted equilibrium measure for some admissible weight $w$. These examples also give new constructions of plurisubharmonic functions in the Lelong class. We also include a list of open problems on the support of extremal measures which are related to solutions of Monge-{A}mpère equations.
Weighted pluripotential theory is a rapidly developing area; and Callaghan \cite{Callaghan} recently introduced $θ$-incomplete polynomials in \cd for $d>1$. In this paper we combine these two theories by defining weighted $θ$-incomplete pluripotential theory. We define weighted $θ$-incomplete extremal functions and obtain a Siciak-Zahariuta type equality in terms of $θ$-incomplete polynomials. Finally we prove that the extremal functions can be recovered using orthonormal polynomials and we demonstrate a result on strong asymptotics of Bergman functions in the spirit of \cite{BermanCn}.