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# Function basis

A particular form of the solution () arises from assuming the existence of a basis function set , such that the function f (x) can be represented by a linear combination of the basis functions in the set, as follows:
 (33)
We can find the linear coefficients ck by multiplying both sides of equation () by one of the basis functions (e.g. ). Inverting the equality
 (34)
where the parentheses denote the dot product, and
 (35)
leads to the following explicit expression for the coefficients ck:
 (36)
Here refers to the kj component of the matrix, which is the inverse of . The matrix is invertible as long as the basis set of functions is linearly independent. In the special case of an orthonormal basis, reduces to the identity matrix:
 (37)

Equation () is a least-squares estimate of the coefficients ck: one can alternatively derive it by minimizing the least-squares norm of the difference between f(x) and the linear decomposition (). For a given set of basis functions, equation () approximates the function f(x) in formula () in the least-squares sense.

Next: Solution Up: Forward interpolation Previous: Interpolation theory
Stanford Exploration Project
12/28/2000