The Error Functional of a Local Cubic-Spline Quadrature Formula for Fourier-Type Integrals: Peano Representation, Optimal Constants and Convergence Classes

The error of the local interpolating cubic-spline quadrature formula for Fourier-type integrals is studied as a bounded linear functional. Its Peano kernel of order three is obtained in closed form. This yields a pointwise estimate that is sharp on functions with bounded third derivative, and a global bound, uniform in the frequency, with constant 0,031399 — almost seven times smaller than the one known before. The optimal constant is identified, the averaged kernel is shown to be positive, decay in the frequency is proved, and convergence is obtained for merely continuous amplitudes.