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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