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Floater–Hormann window polynomials interpolate their local data

Statement

For each window $i$ , the polynomial $p_{i}$ in Local polynomial in a Floater–Hormann window has degree at most $d$ and satisfies $p_{i} (x_{k}) = f_{k}$ for every $i \leq k \leq i + d$ . It is the unique polynomial of degree at most $d$ with those values. In particular, when all local data equal $1$ , the local Lagrange basis functions sum to the constant polynomial $1$ .

Facts & Assumptions

Given: Distinct real nodes and data as in the Floater–Hormann definition, an integer $0 \leq d < n$ , and a window index $0 \leq i \leq n - d$ .

[F1]

Local polynomial in a Floater–Hormann window: $p_{i}$ is the sum of the local Lagrange terms $f_{k} \prod_{j \neq k} (t - x_{j}) / (x_{k} - x_{j})$ .

[F2]

Cardinal-polynomial interpolation and Existence and uniqueness of the cardinal-polynomial interpolant: the displayed local Lagrange sum is the unique polynomial of degree at most $d$ taking the data on the $d + 1$ window nodes.

[F3]

The natural numbers $N$ (von Neumann): every nonnegative integer $d$ belongs to $N$ .

Proof

1.1givenF1algebra

Each summand in [F1] is a polynomial of degree at most $d$ , so $p_{i}$ has degree at most $d$ . At a local node $x_{ℓ}$ , every summand with $k \neq ℓ$ contains the zero factor $x_{ℓ} - x_{ℓ}$ , while the $k = ℓ$ product equals one. Therefore $p_{i} (x_{ℓ}) = f_{ℓ}$ .

2.1givenF2F3step 1.1algebra∎

By [F3], $d \in N$ , so [F2] applies and $p_{i}$ is the unique polynomial of degree at most $d$ with these window values. The constant polynomial $1$ has the value $1$ at each local node; uniqueness makes its Lagrange interpolant equal to $1$ .

Depends on

Used by

Dependency tree · two levels

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Sources

Michael S. Floater and Kai Hormann, Barycentric rational interpolation with no poles and high rates of approximation, Section 2 (standard reference, not scraped)