Unisolvent functions

A collection of n functions f₁, f₂, ..., f_n is unisolvent on domain Ω if the vectors

${\begin{bmatrix}f_{1}(x_{1})\\f_{1}(x_{2})\\\vdots \\f_{1}(x_{n})\end{bmatrix}},{\begin{bmatrix}f_{2}(x_{1})\\f_{2}(x_{2})\\\vdots \\f_{2}(x_{n})\end{bmatrix}},...,{\begin{bmatrix}f_{n}(x_{1})\\f_{n}(x_{2})\\\vdots \\f_{n}(x_{n})\end{bmatrix}}$

are linearly independent for any choice of n distinct points x₁, x₂ ... x_n in Ω. Equivalently, the collection is unisolvent if the matrix F with entries f_i(x_j) has nonzero determinant: det(F) ≠ 0 for any choice of distinct x_j's in Ω.

Unisolvent systems of functions are widely used in interpolation since they guarantee a unique solution to the interpolation problem. Polynomials are unisolvent by the Unisolvence theorem

Examples:

• 1, x, x^2 is unisolvent on any interval by the unisolvence theorem
• 1, x^2 is unisolvent on [0,1], but not unisolvent on [-1,1]
• 1, cos(x), cos(2x), ..., cos(nx), sin(x), sin(2x), ..., sin(nx) is unisolvent on [-pi,pi]

Systems of unisolvent functions are much more common in 1-dimension than in higher dimensions. In dimension d = 2+ (Ω ⊂ R^d), the functions f₁, f₂, ..., f_n cannot be unisolvent on if there exists a single open set on which they are all continuous.

References

Davis: Interpolation and Approximation p. 31-32