The Extremal Truncated Moment Problem

For a degree 2n real d-dimensional multisequence to have a representing measure μ, it is necessary for the associated moment matrix to be positive semidefinite and for the algebraic variety associated to β, , to satisfy rank card as well as the following consistency condition: if a polynomial vanishes on , then . We prove that for the extremal case , positivity of and consistency are sufficient for the existence of a (unique, rank -atomic) representing measure. We also show that in the preceding result, consistency cannot always be replaced by recursiveness of . The first-named author’s research was partially supported by NSF Research Grants DMS-0099357 and DMS-0400741. The second-named author’s research was partially supported by NSF Research Grant DMS-0201430 and DMS-0457138.     

Solution of the Truncated Parabolic Moment Problem

Given real numbers with ₀₀ >0 , the truncated parabolic moment problem for entails finding necessary and sufficient conditions for the existence of a positive Borel measure , supported in the parabola p(x, y) = 0, such that We prove that admits a representing measure (as above) if and only if the associated moment matrix is positive semidefinite, recursively generated and has a column relation p(X, Y) = 0, and the algebraic variety () associated to satisfies card In this case, admits a rank -atomic (minimal) representing measure.Submitted: August 25, 2003     

Solution of the Truncated Hyperbolic Moment Problem

Let Q(x, y) = 0 be an hyperbola in the plane. Given real numbers β ≡ β ⁽²ⁿ⁾={ β _ij } _{i,j ≥ 0,i+j ≤ 2n} , with β₀₀ > 0, the truncated Q-hyperbolic moment problem for β entails finding necessary and sufficient conditions for the existence of a positive Borel measure μ, supported in Q(x, y) = 0, such that We prove that β admits a Q-representing measure μ (as above) if and only if the associated moment matrix is positive semidefinite, recursively generated, has a column relation Q(X,Y) = 0, and the algebraic variety associated to β satisfies card In this case, if then β admits a rank -atomic (minimal) Q-representing measure; if then β admits a Q-representing measure μ satisfying      

Recursively generated weighted shifts and the subnormal completion problem,II

Both authors were partially supported by grants from NSF Raúl Curto was partially supported by a University of Iowa faculty scholar award     

Recursively generated weighted shifts and the subnormal completion problem

Both authors were partially supported by grants from NSF Raúl Curto was partially supported by a University of Iowa faculty scholar award     

Addendum to “The Extremal Truncated Moment Problem”

Due to a technical problem, we accidentally omitted a paragraph from the proof of one of the main results in [1]. In this Addendum we provide the portion of the proof that did not appear in [1]. This is an addendum to .     

The maximum principle for the three chains completion problem

A time-variant version of the maximum principle for the central solution in the commutant lifting theorem is given. The main result is illustrated on the Parrott completion problem.     

On a class of extremal solutions of a moment problem for rational matrix-valued functions in the nondegenerate case II

The main theme of this paper is the discussion of a family of extremal solutions of a finite moment problem for rational matrix functions in the nondegenerate case. We will point out that each member of this family is extremal in several directions. Thereby, the investigations below continue the studies in Fritzsche et al. (in press) [1]. In doing so, an application of the theory of orthogonal rational matrix functions with respect to a nonnegative Hermitian matrix Borel measure on the unit circle is used to get some insights into the structure of the extremal solutions in question. In particular, we explain characterizations of these solutions in the whole solution set in terms of orthogonal rational matrix functions. We will also show that the associated Riesz-Herglotz transform of such a particular solution admits specific representations, where orthogonal rational matrix functions are involved.     

The moment problem on the Wiener space

Consider an L¹-continuous functional ? on the vector space of polynomials of Brownian motion at given times, suppose ? commutes with the quadratic variation in a natural sense, and consider a finite set of polynomials of Brownian motion at rational times, , mapping the Wiener space to R.In the spirit of Schmüdgen's solution to the finite-dimensional moment problem, we give sufficient conditions under which ? can be written in the form ∫⋅dμ for some probability measure μ on the Wiener space such that μ-almost surely, all the random variables are nonnegative.     

On maximum entropy interpolants and maximum determinant completions of associated Pick matrices

In this paper we shall show that the value of the maximum -entropy solution of an interpolation problem of the Nevanlinna-Pick type at the point maximizes the determinant of the solution of an associated matrix completion problem. This serves to show that the solutions of two distinct extremal problems coincide.Harry Dym wishes to express his thanks to Renee' and Jay Weiss for endowing the chair which supports his research.     

Extensions with positive real part. A new version of the abstract band method with applications

This paper presents a new version of the abstract band method. The new scheme applies to extension problems for classes of essentially bounded functions, continuous functions, and bounded operators, which were not covered by earlier versions of the abstract band method.     

Tangential Nevanlinna-Pick Interpolation and Its Connection with Hamburger Matrix Moment Problem

This paper is concerned with the solution of a certain tangential Nevanlinna-Pick interpolation for Nevanlinna functions. We use the so-called block Hankel vector method to establish two intrinsic connections between the tangential Nevanlinna-Pick interpolation in the Nevanlinna class and the truncated Hamburger matrix moment problem associated with the block Hankel vector under consideration: one is a congruent relationship between their information matrices, and the other is a divisor-remainder connection between their solutions. These investigations generalize our previous work on the Nevanlinna-Pick interpolation and power matrix moment problem.     

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 function-theoretic approach to a parametrization of the set of solutions of a completely indeterminate matrical Nehari problem

In this paper, a function-theoretic approach to the completely indeterminate matricial Nehari Problem is given. The uniqueA-normalized resolvent matrixU is constructed by a limit procedure. Moreover, some limit relations for the Potapov-Ginzburg transform ofU are obtained.     

The commutant of rationally cyclic subnormal operators and rational approximation

Let denote the set of analytic bounded point evaluations forR ^q (K, ). Assume that . In this paper, we first show that if is a finitely connected domain and if the evaluation map fromR ^q (K, )L () toH () is surjective, then | is absolutely continuous with respect to harmonic measure for . This generalizes Olin and Yang's corresponding result for polynomials and the proof we present here is simpler. We also provide an example that shows this absolute continuity property fails in general when is an infinitely connected domain. In the second part, we then offer a solution to a problem of Conway and Elias.     

The Schur algorithm for an indefinite Stieltjes moment problem

The nondegenerate truncated indefinite Stieltjes moment problem in the class of generalized Stieltjes functions is considered. To describe the set of solutions of this problem we apply the Schur step‐by‐step algorithm, which leads to the expansion of these solutions in generalized Stieltjes continuous fractions studied recently in [11]. Explicit formula for the resolvent matrix in terms of generalized Stieltjes polynomials is found.