On the finite rank and finite-dimensional representation of bounded semi-infinite Hankel operators

Bounded, semi-infinite Hankel matrices of finite rank over the space ℓ² of square-summable sequences occur frequently in classical analysis and engineering applications. The notion of finite rank often appears under different contexts and the literature is diverse. The first part of this paper reviews some elegant, classical criteria and establishes connections among the various characterizations of finite rank in terms of rational functions, recursion, matrix factorizations and sinusoidal signals. All criteria require 2d parameters, though with different meanings, for a matrix of rank d. The Vandermonde factorization, in particular, permits immediately a singular-value preserving, finite-dimensional representation of the original semi-infinite Hankel matrix and, hence, makes it possible to retrieve the nonzero singular values of the semi-infinite Hankel matrix. The second part of this paper proposes using the LDL∗ decomposition of a specially constructed sample matrix to find the unitarily equivalent finite-dimensional representation. This approach enjoys several advantages, including the ease of computation by avoiding infinite-dimensional vectors, the ability to reveal rank deficiency and the established pivoting strategy for stability. No error analysis is given, but several computational issues are discussed.