Maltsev equal-norm tight frames

Abstract

A frame in $\mathbb{R}^d$ is a set of $n\geqslant d$ vectors whose linear span coincides with $\mathbb{R}^d$. A frame is said to be equal-norm if the norms of all its vectors are equal. Tight frames enable one to represent vectors in $\mathbb{R}^d$ in the form closest to the representation in an orthonormal basis. Every equal-norm tight frame is a useful tool for constructing efficient computational algorithms. The construction of such frames in $\mathbb{C}^d$ uses the matrix of the discrete Fourier transform, and the first constructions of equal-norm tight frames in $\mathbb{R}^d$ appeared only at the beginning of the 21st century. The present paper shows that Maltsev's note of 1947 was decades ahead of its time and turned out to be missed by the experts in frame theory, and Maltsev should be credited for the world's first design of an equal-norm tight frame in $\mathbb{R}^d$. Our main purpose is to show the historical significance of Maltsev's discovery. We consider his paper from the point of view of the modern theory of frames in finite-dimensional spaces. Using the Naimark projectors and other operator methods, we study important frame-theoretic properties of the Maltsev construction, such as the equality of moduli of pairwise scalar products (equiangularity) and the presence of full spark, that is, the linear independence of any subset of $d$ vectors in the frame.