** Next:** Solution
** Up:** Forward interpolation
** Previous:** Interpolation theory

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 *c*_{k} 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
*c*_{k}:
| |
(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
*c*_{k}: 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