On a new positive extension problem for block Toeplitz matrices

We consider positive extension problems for block Toeplitz matrices when the specified entries form a stalactite type pattern. These problems do not seem in general to be amenable to the classical methods (they correspond to bitangential interpolation problems in the class of Carathéodory functions of a kind usually not solvable by the classical methods of interpolation). We solve these problems by reduction to a tangential interpolation with symmetries in a Carathéodory class.