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Least-squares polynomial in an orthogonal-polynomial basis

For orthogonal polynomials

Q_{0}, \dots, Q_{n}

under a positive weighted integral inner product, the least-squares approximation to

f

of degree at most

n

is

p_{n}^{*} = \sum_{k = 0}^{n} \frac{⟨ f , Q _{k} ⟩}{⟨ Q _{k} , Q _{k} ⟩} Q_{k} .

Its residual is orthogonal to every polynomial of degree at most

n

.

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