= Polynomial least-squares approximation
{title2=$p_n=\operatorname*{argmin}_{p\in\Pi_n}\int w(x)|u(x)-p(x)|^2dx$}
= Least-squares polynomial approximation
{synonym}
= Least-squares polynomial
{synonym}
For a positive-definite weighted <inner product> on a compact interval, a <least-squares polynomial> is the <orthogonal projection> of a function onto the finite-dimensional space $\Pi_n$ of <polynomials> of degree at most $n$. In a <basis> $\phi_0,\ldots,\phi_n$, its coefficients solve the <normal equations> $Gc=d$, where $G_{ij}=(\phi_i,\phi_j)$ and $d_i=(u,\phi_i)$. The <Gram matrix> is positive definite: $z^TGz=\|\sum z_i\phi_i\|^2>0$ for $z\ne0$. Thus the coefficients exist and are unique. The residual is <orthogonal> to every candidate polynomial, giving $\|u-q\|^2=\|u-p_n\|^2+\|p_n-q\|^2$ and proving optimality. An orthogonal basis makes these equations diagonal, as in a <least-squares polynomial in an orthogonal-polynomial basis>.
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