There are subtle differences for the shifted Legendre polynomials that prevent them from being used to find a set of vectors that are orthogonal to one another. So one has to find a specific set of functions represented by the matrix A that works for the problem to be fitted. Finding the matrix A is just as difficult as finding the inverse of the correlation matrix so using the matrix of the powers of xk appears to be the simplest procedure.
One needs to see if computation errors affect the results. Usually one can check this by plotting the difference between the fitted function and the calculated or observed function. The error should look something like the next order orthogonal polynomial.
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