Visnyk of V. N. Karazin Kharkiv National University. Ser. Mathematics, Applied Mathematics and Mechanics

The Nevanlinna matrix of the truncated Hausdorff matrix moment problem via orthogonal matrix polynomials on [a,b] for the case of an even number of moments

2026-01-08T11:52:36+00:00

The scalar moment problem was first introduced by T. J. Stieltjes in his work ``Recherches sur les fractions continues'' Annals of the Faculty of Sciences of Toulouse 8, 1--122, (1895). He formulated it as follows: Given the moments of order $k$ ($k=0,1,2,\dots$), find a positive mass distribution on the half-line $[0,+\infty)$.

The study of matrix and operator moment problems was initiated by M. G. Krein in his seminal paper

 ``Fundamental aspects of the representation theory of Hermitian operators with deficiency

 index $(m,m)$'' Translations of the American Mathematical Society, Series II, 97, 75--143, (1949).

This paper is related to the truncated Hausdorff matrix moment (THMM) problem: the truncated moment problem on a compact interval $[a,b]$ in contrast to the Stieltjes moment problem on $[0,+\infty)$ and the Hamburger moment problem on $(-\infty,+\infty)$. Our approach relies on V. P. Potapov’s method, which reformulates interpolation and moment problems as equivalent matrix inequalities and introduces auxiliary matrices that satisfy the $\widetilde{J}_q$--inner function property of the Potapov class, together with a system of column pairs.


The method begins by constructing Hankel matrices from the prescribed moments. If these matrices are positive semidefinite, the THMM problem is solvable. In the strictly positive definite case, known as the non-degenerate case, we transform the associated matrix inequalities to derive the Nevanlinna (or resolvent) matrix of the THMM problem, which characterizes its solutions.

This framework has been extensively applied, for instance in A. E. Choque Rivero, Yu. M. Dyukarev, B. Fritzsche, and B. Kirstein, ``A truncated matricial moment problem on a finite interval'', in Interpolation, Schur Functions and Moment Problems, Operator Theory: Advances and Applications, Birkh\"{a}user, Basel, 165, 121--173, (2006).

The main contribution of the present work is to represent the Nevanlinna matrix of the THMM problem in terms of orthogonal matrix polynomials (OMP) and their associated polynomials of the second kind at point $b$. Note that the representation at point $a$ was obtained earlier in A. E. Choque Rivero, ``From the Potapov to the Krein–Nudel’man representation of the resolvent matrix of the truncated Hausdorff matrix moment problem'' Bulletin of the Mexican Mathematical Society, 21(2), 233--259 (2015).

In addition, we establish new identities involving OMP and reformulate an explicit relationship between the Nevanlinna matrices of the THMM problem at points $a$ and $b$, through OMP.

The Jacobi operator and the stability of vertical minimal surfaces in the sub-Riemannian Lie group SL(2,R)

2026-01-09T11:01:27+00:00

We consider oriented immersed minimal surfaces in three-dimensional sub-Riemannian manifolds which are vertical, i.e., perpendicular to the two-dimensional horizontal distribution of the sub-Riemannian structure. We showed earlier that a vertical surface is minimal in the sub-Riemannian sense if and only if it is minimal in the Riemannian sense and that its sub-Riemannian stability implies its Riemannian stability. We introduce the sub-Riemannian version of the Jacobi operator for such surfaces and prove a sufficient condition for the stability of vertical minimal surfaces similar to a theorem of Fischer-Colbrie and Schoen: if a surface allows a positive function with the vanishing Jacobi operator then it is stable.

Next, we use the Jacobi operator technique to investigate vertical minimal surfaces in the Lie group $\widetilde{\mathrm{SL}(2,\mathbb{R})}$ that can be described as the universal covering of the unit tangent bundle of the hyperbolic plane with the standard left-invariant Sasaki metric (that corresponds to one of the Thurston geometries) and with two different types of sub-Riemannian structures. First, we consider a family of non-left-invariant structures defined by some parameters, find the values of parameters for which vertical minimal surfaces exist, and describe such complete connected surfaces. These are Euclidean half-planes and cylinders, and they all are stable in the sub-Riemannian sense and thus in the Riemannian sense. In particular, this gives us examples of structures that do not allow vertical minimal surfaces. Then, we describe complete connected vertical minimal surfaces for another sub-Riemannian structure that is left-invariant. These are half-planes and helicoidal surfaces that also appear to be stable in the sub-Riemannian sense and thus in the Riemannian sense.

On differentiation with respect to filters

2026-01-07T08:38:18+00:00

The article explores a generalization of the concept of the derivative of a real-valued function of one variable based on filter theory. A new construction is proposed that allows the definition of a derivative of a function with respect to a filter, which reflects the manner in which the variable approaches a given point. Unlike the classical definition, where the limit is taken via a linear approach of the argument, the new definition permits a wider range of approaches to the point, thus providing a more flexible framework for analyzing the local behavior of functions. The introduced concept includes the classical definition of the derivative as a special case when an appropriate filter is chosen. The paper presents proofs of generalized versions of basic derivative properties: linearity, product rule, quotient rule, and chain rule. In particular, it is shown that the derivative with respect to a filter satisfies the same formal differentiation rules as the classical derivative while preserving greater flexibility in how the argument approaches the point. The results obtained expand the scope of differential calculus to cases where the classical approach is either inapplicable or lacks precision or interpretative convenience. It is demonstrated that, in some situations, the derivative with respect to a filter better reflects real processes of change, such as in problems with asymmetric or constrained neighborhoods of a point. The proposed approach opens new perspectives for applications in the theory of generalized functions, measure theory, and functional analysis. The article also provides examples illustrating the application of the new concept and offers a comparative analysis with the classical theory. The presented material may be of interest to researchers in the field of mathematical analysis as well as to educators seeking to extend the traditional approach to differentiation. This work holds both theoretical and methodological value, as it introduces a new tool for further research in the field of modern limit theory.

Closed equivalence relations on compact spaces and pairs of commutative C*-algebras: a Categorical Approach

2026-01-08T19:28:57+00:00

In this paper, we study a categorical extension of the classical Gelfand-Naimark duality between compact Hausdorff spaces and commutative unital C*-algebras. We establish an equivalence between the category of compact Hausdorff spaces with closed equivalence relations and the category of pairs consisting of a commutative unital C*-algebra together with one of its unital C*-subalgebras. The motivation is that Gelfand duality can be enriched by additional structure: closed equivalence relations encode quotient spaces and invariance on the topological side, while subalgebras reflect restrictions and symmetries on the algebraic side. Shilov’s theorem, which identifies closed unital self-adjoint subalgebras of C(X) with algebras of functions invariant under closed equivalence relations, provides an essential link between these settings. We introduce the category EqRel, whose objects are compact Hausdorff spaces with closed equivalence relations and whose morphisms are continuous trajectory-preserving maps, and the category C*Pairs, whose objects are pairs (A,B) with A a commutative unital C*-algebra and B ⊂ A a unital C*-subalgebra, with morphisms given by unital *-homomorphisms preserving B. Contravariant functors are defined in both directions: (X,R) → (C(X),B_R), where B_R consists of functions constant on R-classes, and (A,B) → (Σ(A),R_B), where Σ(A) is the spectrum and R_B relates characters agreeing on B. Using the Kolmogorov-Gelfand theorem, the Gelfand transform, and Shilov’s theorem, we show that these functors are mutually inverse up to morphism of functors and thus prove the categorical equivalence EqRel ≃ C*Pairsop. This result demonstrates that the geometric notion of closed equivalence relations on compact spaces is in perfect correspondence with the algebraic notion of unital subalgebras of commutative C*-algebras.