MERCEDES-BENZ FRAMES IN $$\mathbb {R}^d$$ AS A DIRECT SUM OF A PAIR OF ORTHOGONAL TIGHT FRAMES

Abstract

The well-known Mercedes-Benz frames in $$\mathbb {R}^d$$ are known to exist and they have been constructed too. They are presented here as an external direct sum of a pair of orthogonal frames. A necessary and sufficient condition for a unit norm tight frame to be a Mercedes-Benz frame in $$\mathbb {R}^d$$ is provided in terms of a vector from the null space of its synthesis operator. Also, a necessary and sufficient condition for a unit norm system of vectors to be a Mercedes-Benz frame is provided in terms of the frame potential and a vector from the null space of the synthesis operator.