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

TL;DR

This paper derives explicit formulas for weighted orthogonal polynomials and optimal approximants in the unit 2-ball, using elementary methods that avoid reduction to one dimension, expanding understanding of multivariable polynomial approximation.

Contribution

It provides new closed-form expressions for orthogonal polynomials and optimal approximants in several variables within the unit ball, without relying on one-dimensional reduction.

Findings

Explicit formulas for orthogonal polynomials in the unit 2-ball.

Closed expressions for optimal approximants associated with specific functions.

Elementary proofs that do not depend on reduction to one-dimensional cases.

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.