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Floater–Hormann C^(d+2) parity error bounds

Statement

Let $n \geq 1$ , $0 \leq d < n$ , and $x_{0} < \dots < x_{n}$ be real nodes in $[a, b] = [x_{0}, x_{n}]$ . Put $h = \max_{0 \leq j < n} (x_{j + 1} - x_{j})$ , $m = n - d$ , and $ϵ = 1$ if $m$ is even and $ϵ = 0$ if $m$ is odd. Let $f \in C^{d + 2} ([a, b])$ and let $r_{d}$ be the exact Floater–Hormann interpolant. For a continuous $q$ on $[a, b]$ , write $∥ q ∥_{\infty} = \sup_{x \in [a, b]} ∣ q (x) ∣$ . If $d \geq 1$ , then $∥ f - r_{d} ∥_{\infty} \leq h^{d + 1} (\frac{(b - a) ∥ f^{(d + 2)} ∥_{\infty}}{d + 2} + ϵ \frac{∥ f^{(d + 1)} ∥_{\infty}}{d + 1}) .$ If $d = 0$ , let $β$ be the local mesh ratio in Local mesh ratio for degree-zero Floater–Hormann interpolation. Then $∥ f - r_{0} ∥_{\infty} \leq h (1 + β) (\frac{(b - a) ∥ f^{''} ∥_{\infty}}{2} + ϵ ∥ f^{'} ∥_{\infty}) .$ No quasi-uniformity hypothesis is needed when $d \geq 1$ . If $d = 0$ , a bound $β \leq ρ$ follows from a global mesh-ratio bound $ρ$ by A global mesh ratio bounds the local degree-zero ratio.

Facts & Assumptions

Given: Ordered distinct nodes $x_{0}, \dots, x_{n}$ , an integer $0 \leq d < n$ , $f \in C^{d + 2} ([x_{0}, x_{n}])$ , and the exact Floater–Hormann interpolant.

[F1]

Floater–Hormann weights equal the local-polynomial blend weights and Newton remainder for a Floater–Hormann window polynomial: off the nodes, $f (x) - r_{d} (x) = (\sum_{i = 0}^{n - d} (- 1)^{i} f [x_{i}, \dots, x_{i + d}, x]) / (\sum_{i} λ_{i} (x))$ .

[F2]

Adjacent-window divided-difference recurrence: adjacent terms satisfy $G_{i} - G_{i + 1} = - (x_{i + d + 1} - x_{i}) f [x_{i}, \dots, x_{i + d + 1}, x]$ .

[F3]

Rolle bound for a divided difference: for $f \in C^{q}$ , an order- $q$ divided difference has magnitude at most $∥ f^{(q)} ∥_{\infty} / q!$ .

[F4]

Floater–Hormann barycentric denominator lower bound: on each open gap the denominator has magnitude at least $1 / (d! h^{d + 1})$ for $d \geq 1$ , and at least $1 / [h (1 + β)]$ for $d = 0$ .

[F5]

Floater–Hormann interpolants have no real poles: the quotient has no real poles and interpolates the data at every node.

[F6]

Higher derivatives and the classes $C^{k}$ and $C^{\infty}$ : $C^{d + 2}$ means derivatives through order $d + 2$ exist and are continuous on the closed interval, with one-sided endpoint derivatives.

[F7]

The natural numbers $N$ (von Neumann): $N$ contains $0$ and is closed under successor, so $d + 1, d + 2 \in N$ .

[F8]

Extreme value theorem: a continuous real function on a nonempty compact subset of $R$ attains a greatest and a least value: a continuous real function on a nonempty compact interval has a finite attained maximum and minimum.

Proof

1.1givenF1F6F8algebra

Fix a point $x$ in an open node gap. By [F6], the stated smoothness hypothesis supplies the derivatives used below. By [F8], their sup norms are finite. By [F1], the error equals $N (x) / D (x)$ , where $N (x) = \sum_{i = 0}^{m} (- 1)^{i} G_{i}, G_{i} = f [x_{i}, \dots, x_{i + d}, x], D (x) = \sum_{i} λ_{i} (x) .$ This identity follows by writing $f - r_{d} = \sum_{i} λ_{i} (f - p_{i}) / D$ and substituting the window remainder; $λ_{i} \prod_{k = i}^{i + d} (x - x_{k}) = (- 1)^{i}$ .

2.1givenF2F3F7F8step 1.1algebra

Pair the terms of $N$ as $(G_{0} - G_{1}) + (G_{2} - G_{3}) + \dots$ . If $m$ is odd, these pairs exhaust the sum. If $m$ is even, the final term $G_{m}$ remains. By [F2] each pair is $- (x_{i + d + 1} - x_{i}) f [x_{i}, \dots, x_{i + d + 1}, x]$ . The order- $(d + 2)$ Rolle bound [F3] bounds its divided difference by $∥ f^{(d + 2)} ∥_{\infty} / (d + 2)!$ . Each interval length $x_{i + d + 1} - x_{i}$ is the sum of $d + 1$ adjacent mesh gaps, and each gap occurs in at most $d + 1$ windows. Hence the sum of lengths over the even indices in the pairing is at most $(d + 1) (b - a)$ .

3.1givenF3F7F8step 2.1algebra

If $m$ is odd, step 2.1 gives $∣ N ∣ \leq (d + 1) (b - a) ∥ f^{(d + 2)} ∥_{\infty} / (d + 2)!$ . If $m$ is even, it gives the same bound plus $∣ G_{m} ∣ \leq ∥ f^{(d + 1)} ∥_{\infty} / (d + 1)!$ by [F3] and [F7]. Thus both cases are $∣ N ∣ \leq \frac{(d + 1) (b - a) ∥ f^{(d + 2)} ∥_{\infty}}{(d + 2)!} + ϵ \frac{∥ f^{(d + 1)} ∥_{\infty}}{(d + 1)!} .$ This bounds the numerator in either parity case.

4.1givenF1F4step 3.1algebra

If $d \geq 1$ , divide the numerator bound in step 3.1 by $∣ D ∣ \geq 1 / (d! h^{d + 1})$ from [F4]. Simplifying factorials gives $∣ f (x) - r_{d} (x) ∣ \leq h^{d + 1} (\frac{(b - a) ∥ f^{(d + 2)} ∥_{\infty}}{d + 2} + ϵ \frac{∥ f^{(d + 1)} ∥_{\infty}}{d + 1}) .$ This is the required open-gap bound for $d \geq 1$ .

4.2givenF1F4step 3.1algebra

If $d = 0$ , the same numerator bound in step 3.1, with $∣ D ∣ \geq 1 / [h (1 + β)]$ from [F4], gives $∣ f (x) - r_{0} (x) ∣ \leq h (1 + β) (\frac{(b - a) ∥ f^{''} ∥_{\infty}}{2} + ϵ ∥ f^{'} ∥_{\infty}) .$ This is the required open-gap bound for $d = 0$ .

5.1givenF5F8step 4.1step 4.2∎

At every node the error is zero by [F5]. The no-real-pole property makes $r_{d}$ continuous on the compact interval, so the pointwise estimates on the open gaps extend to the supremum over all $[a, b]$ ; finiteness of that supremum follows from [F8]. The two cases prove the stated bounds.

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Sources

Michael S. Floater and Kai Hormann, Barycentric rational interpolation with no poles and high rates of approximation, Theorems 2–3 (standard reference, not scraped)