Unisolvent functions

In mathematics, a collection of n functions ƒ₁, ƒ₂, ..., ƒ_n is unisolvent on domain Ω if the vectors

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

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 ƒ_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, x² is unisolvent on any interval by the unisolvence theorem
• 1, x² 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 [−π, π]

Systems of unisolvent functions are much more common in 1 dimension than in higher dimensions. In dimension d = 2 and higher (Ω ⊂ R^d), the functions ƒ₁, ƒ₂, ..., ƒ_n cannot be unisolvent on Ω if there exists a single open set on which they are all continuous. To see this, consider moving points x₁ and x₂ along continuous paths in the open set until they have switched positions, such that x₁ and x₂ never intersect each other or any of the other x_i. The determinant of the resulting system (with x₁ and x₂ swapped) is the negative of the determinant of the initial system. Since the functions ƒ_i are continuous, the intermediate value theorem implies that some intermediate configuration has determinant zero, hence the functions cannot be unisolvent.

References

• Philip J. Davis: Interpolation and Approximation pp. 31–32