Interpolation by solutions to linear differential equations (Q354809)

The author considers linear operators with constant coefficients of the form \[ D_{n}(u)=u^{(n)}+a_{1}u^{(n-1)}+a_{2}u^{(n-2)}+...+a_{n}u. \] Let \(\lambda_{k} \) be the characteristic numbers of this operator. Let \( \mathcal{L}_{n} \) denote the set of all solutions of the homogeneous equation \[ D_{n}(u)=0. \] Then \( \mathcal{L}_{n} \) is an \(n\)-dimensional subspace of \(C[a,b]\). It is known [the author, J. Math. Sci., New York 142, No. 1, 1763--1768 (2007); translation from Probl. Mat. Anal. 34, 35--38 (2006; Zbl 1202.41024)] that \( \mathcal{L}_{n} \) is a Chebyshev space for any \([a,b]\) if all the characteristic numbers \(\lambda_{k}\) are real and for \(b-a<\frac{\pi}{(\max\text{Im}\lambda_{j})}\) if there are complex characteristic numbers. In this paper, it is shown that \( \mathcal{L}_{n} \) is an \( ET\)-subspace under the same conditions. It may be noted that an \(n\)-dimensional subspace \( C_{n}\subset C[a,b] \) is called an \( ET\)-subspace if it consists of \(n\)-times continuously differentiable functions and every function of this subspace that is not identically equal to zero has at most \( (n-1)\)-roots, where the multiplicity is taken into account. Further, a representation for the remainder (error) in the Hermitian interpolation of a function by functions in this space is obtained. This result generalizes representation of the remainder in the algebraic interpolation. An example is given to illustrate the results obtained.