Abstract. In a seminal paper, Sarason generalized some classical interpolation problems for H ∞ functions on the unit disc to problems concerning lifting onto H 2 of an operator T that is defined on K = H 2 ⊖φH 2 (φ is an inner function) and commutes with the (compressed) shift S. Inparticular,heshowed that interpolants (i.e., f ∈ H ∞ such that f(S)=T) having norm equal to �T � exist, and that in certain cases such an f is unique and can be expressed as a fraction f = b/a with a, b ∈ K. In this paper, we study interpolants that are such fractions of K functions and are bounded in norm by 1 (assuming that �T � < 1, in which case they always exist). We parameterize the collection of all such pairs (a, b) ∈ K × K and show that each interpolant of this type can be determined as the unique minimum of a convex functional. Our motivation stems from the relevance of classical interpolation to circuit theory, systems theory, and signal processing, where φ is typically a finite Blaschke product, and where the quotient representation is a physically meaningful complexity constraint. 1.

Abstract—We introduce a Kullback–Leibler-type distance between spectral density functions of stationary stochastic processes and solve the problem of optimal approximation of a given spectral density 9 by one that is consistent with prescribed second-order statistics. In general, such statistics are expressed as the state covariance of a linear filter driven by a stochastic process whose spectral density is sought. In this context, we show i) that there is a unique spectral density 8 which minimizes this Kullback–Leibler distance, ii) that this optimal approximate is of the form 9 where the “correction term ” is a rational spectral density function, and iii) that the coefficients of can be obtained numerically by solving a suitable convex optimization problem. In the special case where 9=1, the convex functional becomes quadratic and the solution is then specified by linear equations. Index Terms—Approximation of power spectra, cross-entropy minimization, Kullback–Leibler distance, mutual information, optimization, spectral estimation. I.

Abstract—Over the last several years, a new theory of Nevanlinna–Pick interpolation with complexity constraint has been developed for scalar interpolants. In this paper we generalize this theory to the matrix-valued case, also allowing for multiple interpolation points. We parameterize a class of interpolants consisting of “most interpolants ” of no higher degree than the central solution in terms of spectral zeros. This is a complete parameterization, and for each choice of interpolant we provide a convex optimization problem for determining it. This is derived in the context of duality theory of mathematical programming. To solve the convex optimization problem, we employ a homotopy continuation technique previously developed for the scalar case. These results can be applied to many classes of engineering problems, and, to illustrate this, we provide some examples. In particular, we apply our method to a benchmark problem in multivariate robust control. By constructing a controller satisfying all design specifications but having only half the McMillan degree of conventional controllers, we demonstrate the advantage of the proposed method. Index Terms—Complexity constraint, control, matrix-valued Nevanlinna–Pick interpolation, optimization, spectral

Entropy-like functionals on operator algebras have been studied since the pioneering work of von Neumann, Umegaki, Lindblad, and Lieb. The most well-known are the von Neumann entropy I(ρ): = −trace(ρ log ρ) and a generalization of the Kullback-Leibler distance S(ρ||σ): = trace(ρ log ρ − ρ log σ), refered to as quantum relative entropy and used to quantify distance between states of a quantum system. The purpose of this paper is to explore I and S as regularizing functionals in seeking solutions to multi-variable and multi-dimensional moment problems. It will be shown that extrema can be effectively constructed via a suitable homotopy. The homotopy approach leads naturally to a further generalization and a description of all the solutions to such moment problems. This is accomplished by a renormalization of a Riemannian metric induced by entropy functionals. As an application we discuss the inverse problem of describing power spectra which are consistent with second-order statistics, which has been the main motivation behind the present work.

This paper is dedicated to Giorgio Picci on the occasion of his 65th birthday. I have come to appreciate Giorgio not only as a great friend but also as a great scholar. When we first met at Brown University in 1973, he introduced me to his seminal paper [29] on splitting subspaces, which became the impetus for our joint work on the geometric theory of linear stochastic systems [23–26]. This led to a life-long friendship and a book project that never seemed to converge, but now is close to being finished [27]. I have learned a lot from Giorgio. The present paper grew out of a discussion in our book project, when Giorgio taught me about the connections between prediction-error identification and the Kullback-Leibler criterion. These concepts led directly into the recent theory of analytic interpolation with complexity constraint, with which I have been deeply involved in recent times. I shall try to explain these connections in the following paper. 1

Dedicated to the memory of Mark Grigoryevich Krein on the occasion of the 100th anniversary of his birth Abstract. The moment problem matured from its various special forms in the late 19th and early 20th Centuries to a general class of problems that continues to exert profound influence on the development of analysis and its applications to a wide variety of fields. In particular, the theory of systems and control is no exception, where the applications have historically been to circuit theory, optimal control, robust control, signal processing, spectral estimation, stochastic realization theory and the use of the moments of a probability density. Many of these applications are also still works in progress. In this paper, we consider the generalized moment problem, expressed in terms of a basis of a finite-dimensional subspace P of the Banach space C[a, b] and a “positive ” sequences c, but with a new wrinkle inspired by the applications to systems and control. We seek to parameterize solutions which are positive “rational ” measures, in a suitably generalized sense. Our parameterization is given in terms of smooth objects. In particular, the desired solution space arises naturally as a manifold which can be shown to be diffeomorphic to a Euclidean space and which is the domain of some canonically defined functions. The analysis of these functions, and related maps, yields interesting corollaries for the moment problems and its applications, which we compare to those in the recent literature and which play a crucial role in part of our proof. Our techniques are a combination of those drawn from the literature on the generalized moment problem, from the topology of smooth manifolds and maps, and from convex optimization. Key words. Moment problems, interpolation, rational positive measures

Abstract — The moment problem matured from its various special forms in the late 19th and early 20th Centuries to a general class of problems that continues to exert profound influence on the development of analysis and its applications to a wide variety of fields. In particular, the theory of systems and control is no exception, where the applications have historically been to circuit theory, optimal control, robust control, signal processing, spectral estimation, stochastic realization theory and the use of the moments of a probability density. Many of these applications are also still works in progress. In this paper, we consider the generalized moment problem, expressed in terms of a basis of a finite-dimensional subspace P of the Banach space C[a, b] and a “positive ” sequences c, but with a new wrinkle inspired by the applications to systems and control. We seek to parameterize solutions which are positive “rational” measures, in a suitably generalized sense. Our parameterization is given in terms of smooth objects. In particular, the desired solution space arises naturally as a manifold which can be shown to be diffeomorphic to a Euclidean space and which is the domain of some canonically defined functions. Moreover, on these spaces one can derive natural convex optimization criteria which characterize solutions to this new class of moment problems. I.

This paper is dedicated to Arthur Krener -- a great researcher, a great teacher and a great friend -- on the occasion of his 60th birthday. In this work we study the generalized moment problem with complexity constraints in the case where the actual values of the moments are uncertain. For example, in spectral estimation the moments correspond to estimates of covariance lags computed from a finite observation record, which inevitably leads to statistical errors, a problem studied earlier by Shankwitz and Georgiou. Our approach is a combination of methods drawn from optimization and the di#erentiable approach to geometry and topology. In particular, we give an intrinsic geometric derivation of the Legendre transform and use it to describe convexity properties of the solution to the generalized moment problems as the moments vary over an arbitrary compact convex set of possible values. This is also interpreted in terms of minimizing the Kullback-Leibler divergence for the generalized moment problem