Optimal approximants and orthogonal polynomials in several variables

Abstract

We discuss the notion of optimal polynomial approximants in multivariable reproducing kernel Hilbert spaces. In particular, we analyze difficulties that arise in the multivariable case which are not present in one variable, for example, a more complicated relationship between optimal approximants and orthogonal polynomials in weighted spaces. Weakly inner functions, whose optimal approximants are all constant, provide extreme cases where nontrivial orthogonal polynomials cannot be recovered from the optimal approximants. Concrete examples are presented to illustrate the general theory and are used to disprove certain natural conjectures regarding zeros of optimal approximants in several variables.

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

“…This note continues recent work in [12] concerning certain families of polynomials connected with approximation in spaces of analytic functions, and orthogonal polynomials in weighted spaces. In the paper [12], we discussed the notion of optimal approximants to 1/f for a holomorphic function f belonging to a Hilbert function space in C n , and pointed out connections with orthogonal polynomials in certain weighted spaces, with weight determined by the same target function f .…”

“…This note continues recent work in [12] concerning certain families of polynomials connected with approximation in spaces of analytic functions, and orthogonal polynomials in weighted spaces. In the paper [12], we discussed the notion of optimal approximants to 1/f for a holomorphic function f belonging to a Hilbert function space in C n , and pointed out connections with orthogonal polynomials in certain weighted spaces, with weight determined by the same target function f . We presented some elementary examples of optimal approximants and orthogonal polynomials in several variables, and to obtain concrete closed-form representations of these objects, we relied on one-variable results together with suitable transformations.…”

“…We now state the definition of optimal approximants; see [6,13,2,4,12] for more comprehensive discussions and references. Enumerating the monomials in two variables in some fixed way, we write χ j for the jth monomial in this ordering, and set P n = span{χ j : j = 0, .…”

“…Given some f ∈ H γ , optimal approximants can be viewed as polynomial substitutes for the function 1/f , the point being that 1/f may fall outside of H γ . Optimal approximants arise in several contexts, for instance cyclicity problems and filtering theory, see [13,12]. The papers [8,1,4] discuss some methods for computing optimal approximants, but closed formulas are only known in a few instances.…”

“…There is an important connection between optimal approximants and orthogonal polynomials, as is explained in [3,12]. Namely, if {p * n } denote the optimal approximants to 1/f , f ∈ H γ , and {φ n } are orthogonal polynomials in the weighted space H γ,f , respectively, then…”