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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:
![\begin{displaymath}
f (x) = \sum_{k \in K} c_k \psi_k (x)\;.
\end{displaymath}](img25.png) |
(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
![\begin{displaymath}
\left( \psi_j (x), f (x)\right) = \sum_{k \in K} c_k \Psi_{jk}\;,
\end{displaymath}](img28.png) |
(9) |
where the parentheses denote the dot product, and
![\begin{displaymath}
\Psi_{jk} = \left( \psi_j (x), \psi_k (x)\right) \;,
\end{displaymath}](img29.png) |
(10) |
leads to the following explicit expression for the coefficients
:
![\begin{displaymath}
c_k = \sum_{j \in K} \Psi^{-1}_{kj} \left( \psi_j (x), f
(x)\right) \;.
\end{displaymath}](img30.png) |
(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:
![\begin{displaymath}
\Psi_{jk} = \Psi^{-1}_{kj} = \delta_{jk}\;.
\end{displaymath}](img34.png) |
(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