Some approximation problems in the Hilbert spaces of geophysical data

4 Some approximation problems in the Hilbert spaces of geophysical data 

Suppose. 4 is a linear operator and do are the observed data. We have a set of initial models mi, m2, m and the corresponding data  

Suppose that d is the linearly independent set of elements in D (if not we can select from them the linear independent subset). Evidently we can consider the subspace L spanned over n vectors d,. The problem of approximation is to find the vector da e L, closest to the observed data do D. Evidently 

Using the same approach which I outlined in Appendix A, we may write a system of linear equations for the unknown coefficients oti.  

It can be demonstrated that the linear independence of the elements d guarantees that matrix Fj is nonsingular, which means that the solution to (B.4) aj, i = 1,2,. n always exists for any do and is unique. The Gram matrix can be calculated as follows  

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