Unisolvent point set

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

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

• Numerical Methods / Interpolation

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