Hermite-type exponentially fitted interpolation formulas using three unequally spaced nodes

Commun. Korean Math. Soc. 2022 Vol. 37, No. 1, 303-326 https://doi.org/10.4134/CKMS.c210010 Published online September 29, 2021 Printed January 31, 2022

Kyung Joong Kim Korea Aerospace University

Abstract : Our aim is to construct Hermite-type exponentially fitted interpolation formulas that use not only the pointwise values of an $\omega$-dependent function $f$ but also the values of its first derivative at three unequally spaced nodes. The function $f$ is of the form, \begin{equation*} \begin{array}{ccc} f(x) = g_1(x) \cos (\omega x) + g_2(x) \sin (\omega x), \,\, x \in [a, b], \end{array} \end{equation*} where $g_1$ and $g_2$ are smooth enough to be well approximated by polynomials. To achieve such an aim, we first present Hermite-type exponentially fitted interpolation formulas $I_N$ built on the foundation using $N$ unequally spaced nodes. Then the coefficients of $I_N$ are determined by solving a linear system, and some of the properties of these coefficients are obtained. When $N$ is $2$ or $3,$ some results are obtained with respect to the determinant of the coefficient matrix of the linear system which is associated with $I_N.$ For $N=3,$ the errors for $I_N$ are approached theoretically and they are compared numerically with the errors for other interpolation formulas.