Derivation of general mixed interpolation formula (Q1919507)

The author shows that given any function \(f(x)\), \(x > 0\), there exists a unique \(g(x) = au(kx) + bv(kx) + C_0 + C_1x + \dots + C_{n-2} x^{n-2}\) that interpolates \(f\) at the points \(jh\), \(h > 0\), \(j = 0\), \(1,\dots, n\), where \(u\) and \(v\) are the two linearly independent solutions of a suitable second-order ordinary differential equation. The particular case of \(u(kx) = \cos (kx)\) and \(v(kx) = \sin(kx)\) is discussed, too. Several numerical examples are given.