Hermite nodes from two moments

Abstract

The mean and total squared deviation of positive finite coordinates determine a positive lower Hermite node and an upper node bounding all coordinates.

Theorem 1.1 (Positive lower node and coordinate upper bound).

$$\forall k:\mathrm{Nat}\left(\right),\forall x:\mathrm{Fin}\left(k\right)\to \mathrm{Real}\left(\right),(2\le k\wedge (\forall i:\mathrm{Fin}\left(k\right),0<x\left(i\right)))⟹(0<m-r\wedge (\forall i:\mathrm{Fin}\left(k\right),x\left(i\right)\le m+(k-1)r))$$

Proof. Machine-checked in Lean as D5/S3/Analytic/Interpolation/HermiteMomentBounds.hermite_moment_bounds (✓ std3). ∎

Source. Repository-derived.

Commentary.

Let k be a natural number at least two, and let x assign a positive real coordinate to each element of Fin k. Write m for their arithmetic mean, V for their total squared deviation, and r for the nonnegative radius defined below. The lower and upper nodes are m-r and m+(k-1)r.

$m=\frac{{\sum }_{i\in \mathrm{Fin}\left(k\right)}x\left(i\right)}{k},V={\sum }_{i\in \mathrm{Fin}\left(k\right)}(x\left(i\right)-m{)}^2,r=\sqrt{\frac{V}{k(k-1)}}$

Strict positivity gives a sum of squares strictly smaller than the square of the sum, hence V is less than k(k-1)m squared and r is less than m. The centered coordinates sum to zero. Cauchy-Schwarz on all indices other than a chosen index bounds its squared deviation by (k-1) times the remaining squared deviations. Substitution of the radius gives the upper bound, including when V is zero.

References

• Truth anchor: D5/S3/Analytic/Interpolation/HermiteMomentBounds.hermite_moment_bounds