Equiangular tight frames

A survey on equiangular tight frames in the space is presented. Several equivalent definitions of a tight frame are given. The construction of the Mercedes–Benz frame, the well-known example of a tight frame on the plane, is generalized to the space . The existence problems for the Mercedes–Benz systems and other more general equiangular tight frames are discussed. It is shown that the Welch inequality becomes the equality only on equiangular tight frames (if they exist). Necessary and sufficient conditions for the existence of an equiangular tight (n,m)-frame are formulated in terms of the so-called signature matrices. All the main results are completely proved. Bibliography: 37 titles. Illustrations: 3 figures. Translated from Problems in Mathematical Analysis 39 February, 2009, pp. 3–25.