We establish conditions under which the Bessel operator of "infinite order" is defined and continuous in certain spaces of type S. The correct solvability of the nonlocal multipoint (in time) problem is proved for evolutionary equations of parabolic type with the indicated operators and the initial condition in the form of a generalized function of the ultradistribution type. The properties of the fundamental solution of the posed problem and the properties of the Bessel transform of generalized functions from the spaces of type S. convolutions, convolvers, and multipliers are investigated.