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# Function basis

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.    Forward interpolation  Next: Solution Up: Fomel: Forward interpolation Previous: Interpolation theory

2014-02-21