Classical Interpolation Problems Research Articles

Statistical mechanics of interpolation nodes, pluripotential theory and complex geometry

This is mainly a survey, explaining how the probabilistic (statistical mechanical) construction of Kahler-Einstein metrics on compact complex manifolds, introduced in a series of works by the author, naturally arises from classical approximation and interpolation problems in C-n. A fair amount of background material is included. Along the way the results are generalized to the non-compact setting of C-n. This yields a probabilistic construction of Kahler solutions to Einstein's equations in C-n, with cosmological constant -beta, from a gas of interpolation nodes in equilibrium at positive inverse temperature beta. In the infinite temperature limit, beta -> 0, solutions to the Calabi-Yau equation are obtained. In the opposite zero temperature case the results may be interpreted as transcendental analogs of classical asymptotics for orthogonal polynomials, with the inverse temperature beta playing the role of the degree of a polynomial.

Simplex-splines on the Clough–Tocher element

We propose a simplex spline basis for a space of C1-cubics on the Clough–Tocher split on a triangle. The 12 elements of the basis give a nonnegative partition of unity. We derive two Marsden-like identities, three quasi-interpolants with optimal approximation order and prove L∞ stability of the basis. The conditions for C1-junction to neighboring triangles are simple and similar to the C1 conditions for the cubic Bernstein polynomials on a triangulation. The simplex spline basis can also be linked to the Hermite basis to solve the classical interpolation problem on the Clough–Tocher split.

Maxentropic Solutions to a Convex Interpolation Problem Motivated by Utility Theory

Here, we consider the following inverse problem: Determination of an increasing continuous function U ( x ) on an interval [ a , b ] from the knowledge of the integrals ∫ U ( x ) d F X i ( x ) = π i where the X i are random variables taking values on [ a , b ] and π i are given numbers. This is a linear integral equation with discrete data, which can be transformed into a generalized moment problem when U ( x ) is supposed to have a positive derivative, and it becomes a classical interpolation problem if the X i are deterministic. In some cases, e.g., in utility theory in economics, natural growth and convexity constraints are required on the function, which makes the inverse problem more interesting. Not only that, the data may be provided in intervals and/or measured up to an additive error. It is the purpose of this work to show how the standard method of maximum entropy, as well as the method of maximum entropy in the mean, provides an efficient method to deal with these problems.

Subresultants, Sylvester sums and the rational interpolation problem

We present a solution for the classical univariate rational interpolation problem by means of (univariate) subresultants. In the case of Cauchy interpolation (interpolation without multiplicities), we give explicit formulas for the solution in terms of symmetric functions of the input data, generalizing the well-known formulas for Lagrange interpolation. In the case of the osculatory rational interpolation (interpolation with multiplicities), we give determinantal expressions in terms of the input data, making explicit some matrix formulations that can independently be derived from previous results by Beckermann and Labahn.

Rational interpolation and solitons

We consider the general construction scheme for second-order spectral problems, for which the semiclassical approximation is exact. We show that the inverse spectral problem in this case reduces to the classical interpolation problem for meromorphic functions.

Δ h -Appell sequences and related interpolation problem

A determinantal form for Δ h -Appell sequences is proposed and general properties are obtained by using elementary linear algebra tools. As particular cases of Δ h -Appell sequences the sequence of Bernoulli polynomials of second kind and the one of Boole polynomials are considered. A general linear interpolation problem, which generalizes the classical interpolation problem on equidistant points, is proposed. The solution of this problem is expressed by a basis of Δ h -Appell polynomials. Numerical examples which justify theoretical results on the interpolation problem are given.

On the sequences of the Weyl semi-radii associated with matricial Carathéodory and Schur sequences in both nondegenerate and degenerate cases

In this paper we discuss Weyl matrix balls in the context of the matricial versions of the classical interpolation problems named after Carathéodory and Schur. Our particular focus will be on studying the monotonicity of suitably normalized semi-radii of the corresponding Weyl matrix balls. We, furthermore, devote a fair bit of attention to characterizing the case in which equality holds for particular matricial inequalities. Solving these problems will provide us with a new perspective on the role of the central functions for the classes of Carathéodory and Schur.

A new proof of the Alexander-Hirschowitz interpolation theorem

The classical polynomial interpolation problem in several variables can be generalized to the case of points with greater multiplicities. What is known so far is essentially concentrated in the Alexander-Hirschowitz Theorem which says that a general collection of double points in \({\mathbb{P}^r}\) gives independent conditions on the linear system \({\fancyscript{L}}\) of the hypersurfaces of degree d, with a well known list of exceptions. We present a new proof of this theorem which consists in performing degenerations of \({\mathbb{P}^r}\) and analyzing how \({\fancyscript{L}}\) degenerates.

Semantic interpolation

The problem of interpolation is a classical problem in logic. Given a consequence relation |~ and two formulas φ and ψ with φ |~ ψ we try to find a “simple" formula α such that φ |~ α |~ ψ. “Simple" is defined here as “expressed in the common language of φ and ψ". Non-monotonic logics like preferential logics are often a mixture of a non-monotonic part with classical logic. In such cases, it is natural examine also variants of the interpolation problem, like: is there “simple" α such that φ ⊢ α |~ ψ where ⊢ is classical consequence? We translate the interpolation problem from the syntactic level to the semantic level. For example, the classical interpolation problem is now the question whether there is some “simple" model set X such that M(φ) ⫅ X ⫅ M(ψ). We can show that such X always exist for monotonic and antitonic logics. The case of non-monotonic logics is more complicated, there are several variants to consider, and we mostly have only partial results.

Boundary Interpolation with Blaschke Products of Minimal Degree

We study a version of the classical Nevanlinna-Pick interpolation problem for Blaschke products where all interpolation points are located on the boundary of the unit disc. It turns out that all problems fall into three classes which are distinguished by the minimal degree of the interpolating function. The properties of the classes are studied in some detail. In particular it is shown that exactly one class consists of well-posed problems. An algorithm for classifying a given boundary interpolation problem is developed and a procedure for finding a solution with minimal degree for the generic class of regular problems is described.

On an Extension Problem for Contractive Block Hankel Operator Matrices

Abstract The paper deals with an operator extension problem for contractive block Hankel operators which arose in the context of the operator version of the classical Nehari interpolation problem. V.M. Adamjan, D.Z. Arov, and M.G. Krein [5] obtained that the solution set of this operator extension problem is an operator ball. Hereby, they constructed the parameters of this operator ball via a regularization procedure using the corresponding expressions of the first studied nondegenerate case. The main aim of this paper is to derive more explicit formulas for the parameters of this operator ball. Hereby we use Moore-Penrose inverses of bounded linear operators in Hilbert space.

Generalized interpolation in 𝐻^{∞} with a complexity constraint

In a seminal paper, Sarason generalized some classical interpolation problems for H ∞ H^\infty functions on the unit disc to problems concerning lifting onto H 2 H^2 of an operator T T that is defined on K = H 2 ⊖ ϕ H 2 \mathcal {K} =H^2\ominus \phi H^2 ( ϕ \phi is an inner function) and commutes with the (compressed) shift S S . In particular, he showed that interpolants (i.e., f ∈ H ∞ f\in H^\infty such that f ( S ) = T f(S)=T ) having norm equal to ‖ T ‖ \|T\| exist, and that in certain cases such an f f is unique and can be expressed as a fraction f = b / a f=b/a with a , b ∈ K a,b\in \mathcal {K} . In this paper, we study interpolants that are such fractions of K \mathcal {K} functions and are bounded in norm by 1 1 (assuming that ‖ T ‖ > 1 \|T\|>1 , in which case they always exist). We parameterize the collection of all such pairs ( a , b ) ∈ K × K (a,b)\in \mathcal {K}\times \mathcal {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 \phi is typically a finite Blaschke product, and where the quotient representation is a physically meaningful complexity constraint.

A band method approach to a positive expansion problem in a unitary dilation setting

In this paper a positive commuting expansion problem is presented and solved. The problem is set up in the framework of a minimal unitary dilation of a contraction acting on a Hilbert space and includes the Caratheodory and other classical interpolation problems. By combining the geometry of the minimal unitary dilation with state space techniques from system theory, a special solution is constructed. Next using the band method approach and spectral factorizations of this special solution a linear fractional parameterization of all solutions is obtained. Explicit state space formulas and applications to some classical interpolation problems are given.

Piecewise interpolants on matrix lie groups

Here we consider a numerical procedure to interpolate on matrix Lie groups. By using the exponential map and its (1, 1) diagonal Padé approximant, piecewice interpolants may be derived. The approach based on the Padé map has the advantage that the computation of exponentials and logarithms of matrices are reduced. We show that the updating technique proposed by Enright in [1] may be applied when a dense output is required. The application to the numerical solution of a system ODEs on matrix group and to a classical interpolation problem are reported.

Some adaptive control problems which convert to a "classical" problem in several complex variables

We discuss the equivalence of bi-H/sup /spl infin// control problems to certain problems of approximation and interpolation by analytic functions in several complex variables. In bi-H/sup /spl infin// control, the goal is to perform H/sup /spl infin// control design for a plant where part of it is known and a stable subsystem /spl delta/ is not known, i.e. the response at s is P(s, /spl delta/(s)). We assume that once our system is running, we can identify /spl delta/ online. Thus the problem is to design a function K off-line that uses this information to produce a H/sup /spl infin// controller via the formula K(s, /spl delta/(s)). The controller should yield a closed loop system with H/sup /spl infin// gain at most /spl gamma/ no matter which /spl delta/ occurs. This is a frequency domain problem. The article shows how several bi-H/sup /spl infin// control problems convert to two complex variable interpolation problems. These precisely generalize the classical (one complex variable) interpolation (AAK-commutant lifting) problems which lay at the core of H/sup /spl infin// control. These problems are hard, but the last decade has seen substantial success on them in the operator theory community. In the most ideal of bi-H/sup /spl infin// cases these lead to a necessary and sufficient treatment of the control problem.

On an operator approach to interpolation problems for Stieltjes functions

A general interpolation problem for operator-valued Stieltjes functions is studied using V. P. Potapov's method of fundamental matrix inequalities and the method of operator identities. The solvability criterion is established and under certain restrictions the set of all solutions is parametrized in terms of a linear fractional transformation. As applications of a general theory, a number of classical and new interpolation problems are considered.

The Krein Formula for Generalized Resolvents in Degenerated Inner Product Spaces

Let S be a symmetric operator with defect index (1,1) in a Pontryagin space ℋ. The Krein formula establishes a bijective correspondence between the generalized resolvents of S and the set of Nevanlinna functions as parameters. We give an analogue of the Krein formula in the case that ℋ is a degenerated inner product space. The set of parameters is determined by a kernel condition. These results are applied to some classical interpolation problems with singular data.

Discrete time-variant interpolation as classical interpolation with an operator argument

Necessary and sufficient conditions are derived for the existence of solutions to discrete time-variant interpolation problems of Nevanlinna-Pick and Nudelman type. The proofs are based on a reduction scheme which allows one to treat these time-variant interpolation problems as classical interpolation problems for operator-valued functions with operator arguments. The latter ones are solved by using the commutant lifting theorem.

The problem of completing partial matrix functions as a classical interpolation problem

In this paper we consider the problem of completing a {open_quotes}partial{close_quotes}matrix function of the form that is holomorphic inside the unit disc to a {open_quotes}complete{close_quotes} matrix function that is holomorphic inside the unit disc and has a nonnegative real part in that region. This problem can be treated as a generalization of the problem of weighted approximation in a uniform metric of an antianalytic function by analytic functions. This is the so-called classical interpolation problem. The simplest and best known representatives of the class of classical interpolation problems are the Nevanlinna-Pik problem and the step and trigonometric moments problems. The classical interpolation problems are usually discussed within classes of contracting analytic functions, functions with positive real (:or imaginary) parts, or related classes. A distinguishing characteristic of these problems is the fact that a criterion for their solvability is positivity of some kernel (matrix), and the set of solutions is described in terms of a linear fractional transformation. The positivity of some kernel (matrix), and the set of solutions is described in terms of a linear fractional transformation. The term {open_quotes}classical interpolation problem{close_quotes} prompted Akhiezer to name a monograph {open_quotes}The Classical Moments Problem{open_quotes}; this book was introduced to the mathematicans ofmore » the Odessa school about fifteen years ago, and has become a standard text. One of the most effective means of investigating such problems is a method based on the theory of extensions of operators in Hilbert spaces. The goal of the present paper is to include the completion problem in the theory of classical interpolation problems, which we have already reduced to an abstract schema, and to use this schema to provide criteria for solvability of problems on completion and description of solution sets. Of necessity, our presentation does not stand alone.« less

Nonlinear $H^\infty $ Optimization: A Causal Power Series Approach

In this paper, using a power series methodology a design procedure applicable to analytic nonlinear plants is described. The technique used is a generalization of the linear $H^\infty $ theory. In contrast to previous work on this topic ([Indiana J. Math., 36 (1987), pp. 693–709], [Oper. Theory Adv. Appl., 41 (1989), pp. 255–277], [SIAM J. Control Optim., 27 (1989), pp. 842–860] ), the authors are now able to incorporate explicitly a causality constraint into the theory. In fact, it is shown that it is possible to reduce a causal optimal design problem (for nonlinear systems) to a classical interpolation problem solvable by the commutant lifting theorem [Harmonic Analysis of Operators on Hilbert Space, North-Holland, Amsterdam, 1970], [The Commutant Lifitng Approach to Interpolation Problems, Birkhäuser, Boston, 1990].