Unisolvent point set

In mathematics, a finite collection of points $X\subset \mathbb {R} ^{n}$ is often called unisolvent for a space $W$ if any element $w\in W$ is uniquely determined by its values on $X$ .
$X$ is unisolvent for $\Pi _{n}^{m}$ (polynomials in n variables of degree at most m) if there exists a unique polynomial in $\Pi _{n}^{m}$ of lowest possible degree which interpolates the data $X$ .

Simple examples in $\mathbb {R}$ would be the fact that two distinct points determine a line, three points determine a parabola, etc. It is clear that over $\mathbb {R}$ , any collection of k + 1 distinct points will uniquely determine a polynomial of lowest possible degree in $\Pi ^{k}$ .

References

↑ Karvonen, Toni; Oates, Chris J; Sarkka, Simo (2018). "A Bayes-Sard Cubature Method". Advances in Neural Information Processing Systems. 31. Curran Associates, Inc. Definition 2.4 (Unisolvency). Let π be a finite-dimensional linear subspace of real-valued functions on D. A set X = {x1,...,xn} ⊂ D with n ≥ dim(π)is called π-unisolvent if the zero function is the only element in π that vanishes on X.

External links

Numerical Methods / Interpolation

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[1] Karvonen, Toni; Oates, Chris J; Sarkka, Simo (2018). "A Bayes-Sard Cubature Method". Advances in Neural Information Processing Systems. 31. Curran Associates, Inc. Definition 2.4 (Unisolvency). Let π be a finite-dimensional linear subspace of real-valued functions on D. A set X = {x1,...,xn} ⊂ D with n ≥ dim(π)is called π-unisolvent if the zero function is the only element in π that vanishes on X.

Unisolvent point set

See also

References

External links