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The problem can be solved by computing k inner products. In this context, the coefficients j3{ are called generalized Fourier coefficients. It is a matter of consequence that the coefficients /% are determined independently of one another. This means that having computed an approximation in terms of orthogonal functions, we can add terms at only the additional cost of computing the new coefficients. 3. For a specific example, let us consider the problem of expanding a function / on [—TT, TT] in the Fourier series ,„ ^ The inner product is the usual one that generates the continuous 2-norm: 47 48 Afternotes Goes to Graduate School The first thing is to verify that the cosines and sines are indeed orthogonal.

2. The vector b satisfies the normal equations 3. 9) 4. The approximation Xb is the projection of y onto the column space of X. 5. The residual vector y — Xb is orthogonal to the column space of X. 1. Summary of best approximation in an inner-product space. 13. To derive the classical way, note that XTPx = XT. 7) by XT we obtain This is a A; x A; system of linear equations for b. They are called the normal equations. It is worth noting that the normal equations are really a statement that the residual y — Xb must be orthogonal to the column space of X, which is equivalent to saying that XT(y — Xb} = 0.

Even when p is equal to, say, 2, the maximum error in polynomial p is no more than twice the error in the best approximation. html). 23 24 Afternotes Goes to Graduate School we are talking about errors of order, say, 10~5, then a factor of two is not very much. It is important to stress that the V-P ratio is defined entirely in terms of the / and p and can be computed — at least to reasonable accuracy. One way is to evaluate / and p over a fine grid of equally spaced points and use the values to approximate ||/ — p||oo and to determine good points of alteration U.