Toevalsbasis

Toevalsbasis is a term used in functional analysis and numerical analysis to describe a basis of a function space that is defined with reference to evaluation properties at a set of points. The concept is not a universally standardized term, but it is used in contexts where the coordinates of a function are derived from its values at chosen nodes.

Definition and idea: Let X be a set and V a finite-dimensional subspace of functions on X. A set of functions {b1, ..., bn} ⊂ V is called a toevalsbasis for V with respect to a node set {x1, ..., xn} ⊂ X if the evaluation matrix B, with entries B_{k,i} = bi(xk), is invertible. Then any f ∈ V can be expressed uniquely as f = Σ i ci bi, with the coefficient vector c = B^{-1} [f(x1), ..., f(xn)]^T. In this sense, the coordinates of f are determined by its values at the chosen nodes.

Example: The Lagrange basis for polynomials of degree less than n is a classic example. If the nodes x1, ..., xn are given and Li are the Lagrange basis polynomials satisfying Li(xj) = δ_{ij}, then the evaluation matrix is the identity and any polynomial is determined by its values at the nodes.

Construction and properties: The toevalsbasis depends on both the chosen node set and the target function space. Different node choices yield different bases and conditioning. The approach naturally extends to interpolation schemes and finite element methods, where evaluation-based bases facilitate recovering functions from discrete samples.

Applications and related concepts: interpolation, finite element methods, reproducing kernel Hilbert spaces, and Lagrange or nodal bases. See also evaluation functional and basis in linear algebra.