Manuscript Psy03-D@1 · Psy03-D-v1.tex · theorem · 3e18951d

Theorem: Interpolation with a genuine response certificate

Intervention accounting and certified matching › Conditional interpolation · lines 1599–1638 · proof included or linked

Psy03-D@1#thm:interpolation, Psy03-D@1#eq:interpolationerror

Suppose a declared scalar path has response $q (d) \in C^{2}$ on $[d_{0}, d_{1}]$ , with $d_{0} < d_{1}$ , and

d \in [d_{0}, d_{1}] sup \abs q^{''} (d) \leq M .

Let $ℓ (d)$ be the affine interpolant of the exact endpoint responses. Then

\abs q (d) - ℓ (d) \leq \frac{M}{2} (d - d_{0}) (d_{1} - d) \leq \frac{M ( d _{1} - d _{0} ) ^{2}}{8} .

If the endpoint estimates have simultaneous errors at most $r_{0}, r_{1}$ , their estimated interpolant has allowance

\frac{M}{2} (d - d_{0}) (d_{1} - d) + w_{0} r_{0} + w_{1} r_{1},

where

w_{0} = \frac{d _{1} - d}{d _{1} - d _{0}}, w_{1} = \frac{d - d _{0}}{d _{1} - d _{0}} .

Proof. The interpolation error formula follows from the repeated Rolle theorem applied to the function after subtracting its interpolant and a suitable multiple of $(d - d_{0}) (d - d_{1})$ . Its absolute value is bounded by the displayed curvature allowance. The product is maximized at the midpoint. Endpoint estimation errors propagate with nonnegative weights summing to one.

Displayed formulas in this passage

unlabeled · line 1603
$d \in [d_{0}, d_{1}] sup \abs q^{''} (d) \leq M .$
Psy03-D@1#eq:interpolationerror · line 1609
$\abs q (d) - ℓ (d) \leq \frac{M}{2} (d - d_{0}) (d_{1} - d) \leq \frac{M ( d _{1} - d _{0} ) ^{2}}{8} .$
unlabeled · line 1618
$\frac{M}{2} (d - d_{0}) (d_{1} - d) + w_{0} r_{0} + w_{1} r_{1},$
unlabeled · line 1622
$w_{0} = \frac{d _{1} - d}{d _{1} - d _{0}}, w_{1} = \frac{d - d _{0}}{d _{1} - d _{0}} .$

Theorem: Interpolation with a genuine response certificate

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