Quadrature as functional approximation

Let I integrate a real continuous function over $[a,b]$, and let Q be a weighted sum of values at distinct nodes inside that interval. Suppose Q agrees with I for every polynomial of degree at most d. Subtract such a polynomial p from the target: the quadrature error on p vanishes, leaving only the error functional applied to f−p. Bound the integral by interval length times the uniform norm and the weighted samples by the sum of absolute weights times that norm.

Taking the infimum over p gives the displayed best-approximation bound. This argument needs no derivative of f and makes the role of the rule's exactness explicit. Its strength depends on both the approximation space and the weight norm. A derivative-based remainder, when its hypotheses hold, can be sharper for a particular function. The functional argument is valuable because it separates a general approximation property of the target from the stability properties of the integration rule.

If the rule is exact on constants, its weights sum to the interval length. When every weight is nonnegative, their absolute sum equals that same length, simplifying the error factor to twice the length. Nodal value errors bounded by delta then perturb the quadrature by at most delta times the length. Signed weights may have a much larger absolute sum, allowing small measurement or evaluation errors to cancel badly and be amplified in the final number.

Higher polynomial exactness therefore should not be assessed without weight behavior and available data accuracy. A formula can integrate many monomials exactly in symbolic arithmetic and still be a poor way to process noisy values. Conversely, a stable positive-weight rule may be effective even when a crude worst-case bound is conservative. These statements concern absolute error. Relative accuracy becomes delicate when the true integral is near zero because positive and negative parts of the integrand nearly cancel.

For ordinary integration on a finite interval, the m-point Gaussian rule uses the m distinct interior zeros of its degree-m Legendre polynomial. Choose weights by integrating the associated degree-(m−1) Lagrange basis. To see exactness through degree 2m−1, divide a polynomial h of that degree by the orthogonal polynomial: h equals its product with a polynomial of degree at most m−1 plus a remainder of degree at most m−1. Orthogonality makes the product's integral zero, and the product vanishes at every quadrature node.

Both I and Q therefore reduce to the same remainder, which is integrated exactly by construction. The weights are positive: apply the established exactness to the square of each Lagrange basis polynomial, whose degree is at most 2m−2; its integral is positive and its quadrature sum equals that basis weight. This proves a structural advantage, not exactness for every analytic function. A rule's polynomial degree of exactness supplies a benchmark class, while a general target still needs an error analysis.