LOCALLY CONVEX SPACES OF VECTOR-VALUED DISTRIBUTIONS WITH MULTIPLICATIVE STRUCTURES

Abstract

We construct an infinite-dimensional linear space [Formula: see text] of vector-valued distributions (generalized functions), or sequences, f*(x)=(fn(x)) finite from the left (i.e. fn(x)=0 for n<n0(f*)) whose components fn(x) belong to the linear span [Formula: see text] generated by the distributions δ(m-1)(x-ck), P((x-ck)-m), xm-1, where m=1, 2, …, ck ∈ ℝ, k = 1, …, s. The space of distributions [Formula: see text] can be realized as a subspace in [Formula: see text] This linear space [Formula: see text] has the structure of an associative and commutative algebra containing a unity element and free of zero divizors. The Schwartz counterexample does not hold in the algebra [Formula: see text]. Unlike the Colombeau algebra, whose elements have no explicit functional interpretation, elements of the algebra [Formula: see text] are infinite-dimensional Schwartz vector-valued distributions. This construction can be considered as a next step and a "model" on the way of constructing a nonlinear theory of distributions similar to that developed by L. Schwartz. The obtained results can be considerably generalized.