The following problem is discussed. Let f be a generalized function of slow growth with support on the positive semi-axis, and let #phi#_k be a sequence of "test" functions such that #phi#_k → #phi#_0 as k → +∞ in some function space. Assume that the following limit exists: 1/(ρ(k))(f(kt), #phi#_k(t)) → c, k → +∞, where ρ(k) is a regularly varying function. Find conditions under which the limit 1/(ρ(k))(f(kt), #phi#(t)) → c_#phi#, k → +∞, exists for all test functions #phi#. We state and prove theorems that solve this problem and apply them to the problem of existence of quasi-asymptotics for the solution of an ordinary differential equation with variable coefficients. We prove Abelian and Tauberian theorems for a wide class of integral transformations of distributions, for example, the generalized Stieltjes integral transformation.
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