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Bounded variation controls adjacent Hermite–Genocchi windows

Statement

Let $n \geq 1$ , let $0 \leq d < n$ , let $x_{0} < \dots < x_{n}$ be real nodes, and fix a real query point $x \in [x_{0}, x_{n}]$ distinct from the nodes. Let $f \in C^{d + 1} ([x_{0}, x_{n}])$ , put $g = f^{(d + 1)}$ , and suppose $g$ has finite total variation $V = {Var}_{[x_{0}, x_{n}]} (g)$ as defined in Bounded variation and total variation on an interval. For $0 \leq i \leq n - d$ , put $G_{i} = f [x_{i}, \dots, x_{i + d}, x]$ . Then $\sum_{i = 0}^{n - d - 1} ∣ G_{i + 1} - G_{i} ∣ \leq \frac{V}{(d + 1)!} .$

Facts & Assumptions

Given: $n \geq 1$ , $0 \leq d < n$ , ordered real nodes $x_{0} < \dots < x_{n}$ , a distinct query point $x$ in their interval, $f \in C^{d + 1}$ , and $Var (f^{(d + 1)}) = V < \infty$ .

[F1]

Hermite–Genocchi simplex formula for divided differences: an order- $(d + 1)$ divided difference is the integral of $f^{(d + 1)}$ over the standard simplex, whose volume is $1 / (d + 1)!$ .

[F2]

Bounded variation and total variation on an interval: $V$ is the supremum, over finite partitions of the interval, of the sums of absolute successive increments of $g$ .

[F3]

Higher derivatives and the classes $C^{k}$ and $C^{\infty}$ : $C^{d + 1}$ means that the derivatives through order $d + 1$ exist and are continuous on the closed interval.

[F4]

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

Proof

1.1givenF1F3F4construct

By [F4], $d + 1 \in N$ , so apply [F1] to each $G_{i}$ , using the same simplex coordinates for each window and using the final coordinate for the fixed point $x$ . Thus $G_{i} = \int_{Δ_{d + 1}} g (y_{i} (u)) d u, y_{i} (u) = u_{x} x + \sum_{k = 0}^{d} u_{k} x_{i + k},$ where all coefficients are nonnegative and sum to one. By [F3], $g$ is continuous on the closed interval. Therefore every $y_{i} (u)$ lies in $[x_{0}, x_{n}]$ .

2.1givenF1F2step 1.1algebra∎

For each fixed $u$ , the sequence $y_{0} (u), y_{1} (u), \dots, y_{n - d} (u)$ is nondecreasing: each summand $x_{i + k}$ is nondecreasing in $i$ and has coefficient $u_{k} \geq 0$ . Remove repeated values from this finite sequence and insert the interval endpoints; the definition [F2] then gives $\sum_{i = 0}^{n - d - 1} ∣ g (y_{i + 1} (u)) - g (y_{i} (u)) ∣ \leq V .$ Using [F1], the triangle inequality for integrals, and this pointwise bound, $\sum_{i} ∣ G_{i + 1} - G_{i} ∣ \leq \int_{Δ_{d + 1}} \sum_{i} ∣ g (y_{i + 1} (u)) - g (y_{i} (u)) ∣ d u \leq V vol (Δ_{d + 1}) = \frac{V}{(d + 1)!} .$ This is the claimed total variation bound.

Depends on

Used by

Floater–Hormann error bounds for a bounded-variation highest derivative Theorem

Dependency tree · two levels

24 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

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