arXiv subjects
Publications and source records attributed to Alan R. Legg.
We take a look at weighted Szego projections on ellipses and ellipsoids in light of some known results of real and complex potential theory. We show that on planar ellipses there is a weighted Szego projection taking polynomials to polynomials without increasing degree.
We explore the connection between supports of equilibrium measures and quadrature identities, especially in the case of point sources added to the external field $Q(z)=|z|^{2p}$ with $p \in \mathbb{N}$. Along the way, we describe some quadrature domains with respect to weighted area measure $|z|^{2p}dA_z$ and complex boundary measure $|z|^{-2p}dz$.
The best uniform polynomial approximation of the checkmark function $f(x)=|x-α|$ is considered, as $α$ varies in $(-1,1)$. For each fixed degree $n$, the minimax error $E_n (α)$ is shown to be piecewise analytic in $α$. In addition, $E_n(α)$ is shown to feature $n-1$ piecewise linear decreasing/increasing sections, called V-shapes. The points of the alternation set are proven to be monotone increasing in $α$ and their dynamics are completely characterized. We also prove a conjecture of Shekhtman that for odd $n$, $E_n(α)$ has a local maximum at $α=0$.
We make use of the Bergman kernel function to study quadrature domains for square-integrable holomorphic functions of several variables. Emphasis is given to generalizing biholomorphic mapping properties of planar quadrature domains to the several-dimensional setting.
We reveal a complex analogue to a result about polynomial solutions to the Dirichlet Problem on ellipsoids in $\mathbb{R}^n$ by showing that the Bergman projection on any ellipsoid in $\mathbb{C}^n$ is such that the projection of any polynomial function of degree at most $N$ is a holomorphic polynomial function of degree at most $N$. The discussion is motivated by a connection between the Bergman projection and the Khavinson-Shapiro conjecture in $\mathbb{C}$. We also relate the Khavinson-Shapiro conjecture to polyharmonic Bergman projections in $\mathbb{R}^n$ by showing that these projections take polynomials to polynomials on ellipsoids.