Next: Solution
Up: Fomel: Forward interpolation
Previous: Interpolation theory
A particular form of the solution (1) arises from
assuming the existence of a basis function set
, such that the function can be represented by a linear
combination of the basis functions in the set, as follows:
|
(8) |
We can find the linear coefficients by multiplying both
sides of equation (8) by one of the basis functions
(e.g. ). Inverting the equality
|
(9) |
where the parentheses denote the dot product, and
|
(10) |
leads to the following explicit expression for the coefficients
:
|
(11) |
Here
refers to the 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:
|
(12) |
Equation (11) is a least-squares estimate of the coefficients
: one can alternatively derive it by minimizing the least-squares
norm of the difference between and the linear
decomposition (8). For a given set of basis functions,
equation (11) approximates the function in formula
(1) in the least-squares sense.
Next: Solution
Up: Fomel: Forward interpolation
Previous: Interpolation theory
2014-02-21