Maltsev equal-norm tight frames

A frame in $\mathbb{R}^d$ is a set of $n\geqslant d$ vectors whose linear spancoincides with $\mathbb{R}^d$. A frame is said to be equal-norm if the normsof all its vectors are equal. Tight frames enable one to represent vectorsin $\mathbb{R}^d$ in the form closest to the representation in an orthonormalbasis. Every equal-norm tightframe 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 Maltsevconstruction, such as the equality of moduli of pairwise scalar products (equiangularity) and the presence of full spark, that is, the linear independence ofany subset of $d$ vectors in the frame.

matrix, equal-norm tight frame, analysis operator, synthesis operator, orthogonal rows of a matrix, equiangular frame, full spark