Compound invariants and mixed F-, DF-power spaces

The problems on isomorphic classification and quasiequivalence of bases are studied for the class of mixed F-, DF-power series spaces, i.e, the spates of the following kind [GRAPHICS] where a(i)(p, q) = exp((p - lambda(i)q)a(i)), p,q is an element of N, and l ambda = (lambda(i))(i is an element of N), a = (a(i))(i is an element of N) are some sequences of positive numbers. These spaces. up to isomorphisms, are basis subspaces of tensor products of power series spaces of F- and DF- types. respectively. The m-rectangle characteristic mu(m)(lambda,a)(delta, epsilon: tau, t), m is an element of N of the space G(lambda, a) is defined as the number of members of the sequence (lambda(i), a(i))(i is an element of N) which are contained in the union of m rectangles P-k = (delta(k), ep silon(k)] x (tau(k), t(k)], k = 1,2,...,m. It is shown that each m-rectangl e characteristic is an invariant on the considered class under some proper definition of an equivalency relation. The main tool are new compound invar iants, which combine some version of the classical approximative dimensions (Kolmogorov, Pelezynski) with appropriate geometrical and interpolational operations under neighborhoods of the origin (taken from a given basis).