Orthogonal polynomials and projection error

Work on a finite nondegenerate interval $[a,b]$ with an integrable weight w that is positive almost everywhere. Define the real weighted inner product by integrating the product against w. All polynomials have finite norm on this compact interval, and a nonzero polynomial has positive norm because it vanishes at only finitely many points. Applying Gram–Schmidt to the powers of x gives polynomials $p_k$ of degree k that are mutually orthogonal. Their scaling remains a convention, not a mathematical necessity.

Uniform weight on $[-1,1]$ gives the Legendre family; the weight $1/\sqrt{1-x^2}$ gives the Chebyshev family. These are different geometries, so the projection coefficients of the same function can differ. With a finite discrete sampling measure, sufficiently high polynomial degrees can become indistinguishable at the sample points; that setting needs a rank check rather than an automatic infinite orthogonal family. Here positivity almost everywhere avoids that degeneracy and makes each finite polynomial projection well defined.

Subtract the displayed projection from f and take its inner product with any basis polynomial through degree n. The coefficient formula makes each such inner product vanish, so the residual is orthogonal to the whole approximation space. For any competing polynomial q in that space, decompose f−q into the residual f−P_nf plus the polynomial P_nf−q. The two terms are orthogonal and their squared norms add. No competing q can improve the first term, and equality requires q to equal the projection in this positive-definite space.

This is an exact characterization, not an artifact of a particular numerical implementation. A stable algorithm still has to evaluate or estimate the integrals, handle finite precision, and represent the polynomial without unnecessary cancellation. Computing coefficients from noisy samples creates an additional statistical or quadrature problem. Conversely, even perfectly computed coefficients optimize only the selected weighted L² objective. A narrow residual spike may contribute little integrated energy and still be unacceptable under a uniform tolerance.

If the orthonormal polynomials $\phi_k$ form a complete basis of the weighted L² space and $c_k=\langle f,\phi_k\rangle_w$, Parseval gives residual energy equal to the sum of squared coefficients beyond n. Completeness is essential to equality with the entire infinite tail; finite orthogonality alone only gives the finite Pythagorean identities and Bessel's inequality. For the positive integrable weight on our compact interval, standard polynomial density provides the needed completeness.

Rapid coefficient decay makes low-degree compression effective, but the interpretation remains norm-specific. Differentiating a polynomial expansion weights high degrees more strongly and may amplify a tail that was small in L². Point evaluation is also not a bounded functional on general L² equivalence classes, so an L² error estimate cannot simply be read as a pointwise bound. To demand uniform accuracy or derivative accuracy, add appropriate regularity assumptions and use estimates for those stronger objectives.