Optimal Quadrature Formula with Multiple Nodes in a Hilbert Space

Background: Quadrature formulas that use derivative values alongside function values attain a higher order of accuracy for a fixed number of nodes. In the variational setting of Sard the nodes are fixed in advance and only the weights are chosen, so as to minimize the norm of the error functional over a normed space of integrands. Constructions of this kind have been carried out mainly in Sobolev spaces whose seminorm is generated by a pure derivative of order ????, which yields exactness on polynomials only, and the derivative weights have usually been retained at the endpoints of the interval alone. Purpose: The aim of the work is to construct an optimal quadrature formula in the sense of Sard in the Hilbert space ????(3,2)2,????(0, 1) in which the first-derivative weights are retained at every node, while the function-value weights are fixed to those of the trapezoidal rule, so that exactness for the constant, the linear function and a decaying exponential is built into the formula. Method: The construction follows the method of S. L. Sobolev. The extremal function of the error functional is determined, and the Riesz representation theorem then yields a closed-form expression for the square of the norm of that functional as a quadratic function of the unknown derivative weights. Minimizing this quadratic under the exactness constraints by the method of Lagrange multipliers leads to a linear algebraic system for the weights and the multipliers. Results: An explicit representation of the error functional and of the square of its norm is obtained, which gives a sharp upper bound for the error of the formula on the whole space. The resulting system has order ????+3 and is of Wiener–Hopf type; its unique solvability guarantees that the optimal weights are determined unambiguously. Implication: The result extends the known Sard-type constructions to a third-order Hilbert space that combines exponential exactness with derivative weights retained at the interior nodes.