Unisolvent functions


In mathematics, a set of n functions f1, f2,..., fn is unisolvent on a domain Ω if the vectors
are linearly independent for any choice of n distinct points x1, x2... xn in Ω. Equivalently, the collection is unisolvent if the matrix F with entries fi has nonzero determinant: det ≠ 0 for any choice of distinct xj's in Ω. Unisolvency is a property of vector spaces, not just particular sets of functions. That is, a vector space of functions of dimension n is unisolvent if given any basis, the basis is unisolvent. This is because any two bases are related by an invertible matrix, so one basis is unisolvent if and only if any other basis is unisolvent.
Unisolvent systems of functions are widely used in interpolation since they guarantee a unique solution to the interpolation problem. The set of polynomials of degree at most are unisolvent by the unisolvence theorem.

Examples

Systems of unisolvent functions are much more common in 1 dimension than in higher dimensions. In dimension d = 2 and higher, the functions f1, f2,..., fn 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 xi. The determinant of the resulting system is the negative of the determinant of the initial system. Since the functions fi are continuous, the intermediate value theorem implies that some intermediate configuration has determinant zero, hence the functions cannot be unisolvent.