Multiple interpolation in the kernel of the convolution operator

The multipoint de la Vallée Poussin problem (multiple interpolation problem) in the kernel of the convolution operator is considered in terms of sequentially sufficient sets. Conditions are found under which the zero set of the characteristic function of the convolution operator is sequentially sufficient in the kernel of the dual operator, which implies the solvability of the multipoint de la Vall?e-Poussin problem in the kernel of the original convolution operator.