H-distributions with unbounded multipliers

We investigate H-distributions for sequences in the dual pairs of Bessel spaces, and by the use of unbounded multipliers, with the finite regularity, as test functions. The results relating weak convergence, H-distributions and strong convergence are applied in the analysis of strong convergence for a sequence of approximated solutions to a class of differential equations , where P(x, D) is a differential operator of order k with coefficients in the Schwartz class and is a strongly convergent sequence in an appropriate Bessel potential space.