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## Function basis

To obtain the solution (1), let us assume the existence of a function basic , such that the function f (x) can be represented by a linear combination of the basis functions, as follows:
 (7)
The linear coefficients ck can be found by multiplying both sides of equation (7) by one of the basis functions (e.g. ). Inverting the equality
 (8)
where the parentheses denote the dot product, and
 (9)
gives us the following explicit expression for the coefficients ck:
 (10)
Here refers to the kj component of the matrix, inverse to . 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:

Equation (10) is a least-square estimate of the coefficients ck. For a given set of basis functions, it approximates the function f in formula (1) in the least-square sense.

Next: Solution Up: Problem formulation Previous: Problem formulation
Stanford Exploration Project
11/11/1997