Optimal approximants and orthogonal polynomials in several variables II: families of polynomials in the unit ball

Abstract

We obtain closed expressions for weighted orthogonal polynomials and optimal approximants associated with the function $f(z)=1-\frac{1}{\sqrt{2}}(z_1+z_2)$ and a scale of Hilbert function spaces in the unit $2$-ball having reproducing kernel $(1-\langle z,w\rangle)^{-\gamma}$, $\gamma>0$. Our arguments are elementary but do not rely on reduction to the one-dimensional case.