The multiple interpolation de La Vallée Poussin problem

This article is concerned with the solving of multiple interpolation de La Vallée Poussin problem for generalized convolution operator. Particular attention is paid to the proving of the sequential sufficiency of the set of solutions of the generalized convolution operator characteristic equation. In the generalized Bargmann–Fock space the adjoint operator of multiplication by the variable $z$ is the generalized differential operator. Using this operator we introduce the generalized shift and generalized convolution operators. Applying the chain of equivalent assertions we obtain the fact that the multiple interpolation de La Vallée Poussin problem is solvable if and only if the composition of generalized convolution operator with multiplication by the fixed entire function $\psi(z)$ is surjective. Zeros of the function $\psi(z)$ are the nodes of interpolation. The surjectivity of composition of the generalized convolution operator with the multiplication comes down to the proof of the sequential sufficiency of the set of zeros of a generalized convolution operator characteristic function in the set of solutions of the generalized convolution operator with the characteristic function $\psi(z)$. In the proof of the sequential sufficiency it became necessary to consider the relation of eigenfunctions for different values of $\mu_i.$ The eigenfunction with great value of $\mu_i$ tends to infinity faster than eigenfunction with a lower value for $z$ tends to infinity. The derivative of the eigenfunction of higher order tends to infinity faster than lower-order derivatives with the same values of $\mu_i$. A significant role is played by the fact that the kernel of the generalized convolution operator with characteristic function $\psi(z)$ is a finite sum of its eigenfunction and its derivatives. Using the Fischer representation, Dieudonne–Schwartz theorem and Michael's theorem on the existence of a continuous right inverse we obtain that if the zeros of the characteristic function of a generalized convolution operator are located on the positive real axis in order of increasing then multiple interpolation de La Vallée Poussin problem is solvable in the interpolation nodes.