Least Squares Fit and Determination of Chebyshev Series Coefficients

Let us write the truncation error for an order approximation of :

One measure of goodness of fit of a polynomial to a function is the least squares integral

for some weighting function . To minimize , we set for all the , finding

Let the weighting function be . Then

Using the orthogonality relations, we arrive at the results

and

for

These are equivalent to

and

for

where .