Function basis

To obtain the solution (1), let us assume the existence of a function basic $\{\psi_k(x)\},\;k \in K$, such that the function f (x) can be represented by a linear combination of the basis functions, as follows:  

$$f (x) = \sum_{k \in K} c_k \psi_k (x)\;.$$ (7)

The linear coefficients c_k can be found by multiplying both sides of equation (7) by one of the basis functions (e.g. $\psi_j (x)$). Inverting the equality  

$$\left( \psi_j (x), f (x)\right) = \sum_{k \in K} c_k \Psi_{jk}\;,$$ (8)

where the parentheses denote the dot product, and  

$$\Psi_{jk} = \left( \psi_j (x), \psi_k (x)\right) \;,$$ (9)

gives us the following explicit expression for the coefficients c_k:  

$$c_k = \sum_{j \in K} \Psi^{-1}_{kj} \left( \psi_j (x), f (x)\right) \;.$$ (10)

Here $\Psi^{-1}_{kj}$ refers to the kj component of the matrix, inverse to $\Psi$. The matrix $\Psi$ is invertible as long as the basis set of functions is linearly independent. In the special case of an orthonormal basis, $\Psi$ reduces to the identity matrix:

$$\Psi_{jk} = \Psi^{-1}_{kj} = \delta_{jk}\;.$$

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