Special tight frames

Special vector systems, in which every element is the preceding element multiplied by a unitary matrix U, are introduced. Necessary and sufficient conditions for such a system to be a tight frame are obtained (Theorem 1). Examples illustrating the necessity of every condition are given. The theorem is applied to the Mercedes-Benz frame. Let P denote the matrix composed of orthonormal eigenvectors of U. A new system of vectors in which every element equals the corresponding element of the initial system multiplied by P* is considered. It is proved that this system is a generalized harmonic frame if and only if the assumptions of Theorem 1 hold. This result is applied to show how to transform the Mercedes-Benz frame into a generalized harmonic frame.