Both B-splines and elementary minimum splines have the minimum multiplicity of covering by supports of basis splines at a given approximation order. But the former are not an interpolation basis, while the latter are. The former are positive and have the maximum smoothness, while the latter change signs and are continuous at best. In the present paper we introduce a class of A-minimum splines which involves both types of splines mentioned above. We also consider the notion of g-continuity, which generalizes the qualified smoothness, and cite the necessary and sufficient conditions for the introduced splines to be g-continuous. New families of minimum splines are proposed.