Equiangular Tight Frames with Simplices and with Full Spark in $$\boldsymbol{\mathbb{R}}^{\boldsymbol{d}}$$

Abstract

An equiangular tight frame (ETF) is an equal norm tight frame with the same sharp angles between the vectors. This work is an attempt to create a brief review with complete proofs and calculations of two directions of research on the equiangular tight frames (ETF): bounds of the spark of the ETF, namely the smallest number of the vectors from ETF that are linearly dependent, and the existence of a regular simplex inside ETF. Tracing these two directions, we go through the case of equality in the Welch estimate, see the connection between RIP (restricted isometry property) and the spark of an ETF, construct a regular simplex using the technique of Naimark complements. We show the connection between equality in the lower estimate of the spark and the presence of a simplex inside ETF. Gram matrix and the matrix of the synthesis operator are calculated for the frame with 10 elements in the space $$\mathbb{R}^{5}.$$ This frame contains a regular simplex and its spark is equal to 4, i.e. is not full. On the other hand you may see an example of the ETF with 6 vectors in $$\mathbb{R}^{3}$$ without a simplex, but with the spark equal to 4. In such cases the term ‘‘full spark’’ is used, so we have an example of the full spark ETF (FSETF). Finally, there is a proof of the necessary condition for the existence of the FSETF in the space $$\mathbb{R}^{d}$$ with more than $$d+1$$ vectors, namely, the number of vectors in such a frame is necessary equal to $$2d.$$