Optimal error bounds for the derivatives of two-point mixed interpolation (Q1900357)

Let \(x(t)\in C^{(N)} [0,1 ]\), \(N\geq 2\) be a given function, and let \(P_{N-1} (t)\) be one of its ``classical'' interpolating polynomials of degree \((N-1)\). Let \(e(t)= x(t)- P_{N-1} (t)\) be the associated error. The purpose of this paper is to determine optimal/sharp constants \(C_{N,k}\) in the inequalities \[ |e^{(k)} (t)|\leq C_{N,k} \max_{0\leq t\leq 1}|x^{(N)} (t)|, \qquad 0\leq k\leq N-1 \] for different interpolating polynomials. The investigated interpolating polynomials satisfy: (i) \((n,p)\) and \((p,n)\) conditions, which arise in determining the intervals of nonoscillation for the linear ordinary differential equations, (ii) particular cases of two-point Birkhoff's conditions, (iii) two-point Abel-Goncharov-Hermite conditions, or (iv) two-point Abel-Goncharov-Lidstone conditions. In the study of boundary value problems these inequalities play a fundamental role, for example, in determining the rigions of existence and uniqueness of solution.