Trigonometric Hermite interpolation method for Fredholm linear integral equations (Q6085006)

This paper deals with a new Hermite interpolation method, based on uniform algebraic trigonometric quadratic B-splines, on a bounded interval \([a,b]\) of \(\mathbb R\). It defines a Hermite operator, reproducing the space of algebraic trigonometric functions, generated by \(\{1,\cos x,\sin x\}\), providing an optimal approximation order. Moreover a Hermite basis for such a space is obtained by imposing interpolation conditions for the function \(f\) and its first derivative at the partition knots in \([a,b]\). Despite the major disadvantage of instability, due to the non-positivity of some of the basis elements, such an operator has been successfully used to define a quadrature formula and numerical results, compared with some known ones in literature, confirm this performance. Also the application of such a new quadrature formula to find the numerical solution of Fredholm linear integral equations provides very good results.