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 − 1 2 ( z 1 + z 2 ) f(z)=1-\frac {1}{\sqrt {2}}(z_1+z_2) and a scale of Hilbert function spaces in the unit 2 2 -ball having reproducing kernel ( 1 − ⟨ z , w ⟩ ) − γ (1-\langle z,w\rangle )^{-\gamma } , γ > 0 \gamma >0 . Our arguments are elementary but do not rely on reduction to the one-dimensional case.

Optimal approximants and orthogonal polynomials in several variables

“…, including a full determination of H 2 2 -optimal approximants and orthogonal polynomials, is presented in [35].…”

A note on cyclic vectors in Dirichlet-type spaces in the unit ball of

“…Also, non-zero constant functions are cyclic in every space D α (B n ). More information regarding cyclic vectors in Dirichlet-type spaces over the disk, the polydisc and the unit ball can be found in [3], [4], [5], [6], [8], [12], [13], [14], [20], [21].…”

“…Norm comparisons, sharp decay of norms for special subspaces, capacity conditions studied in [3, 4, 6, 21] are the main motivation for this work. The cyclicity of a function f in a space of holomorphic functions is connected also with the problem of approximating (see [19, 20] for the study of this subject).…”