Mathematics

We prove that if f and g are holomorphic functions on an open connected domain, with the same moduli on two intersecting segments, then f = g up to the multiplication of a unimodular constant, provided the segments make an angle that is an irrational multiple of π . We also prove that if f and g are functions in the Nevanlinna class, and if |f | = |g| on the unit circle and on a circle inside the unit disc, then f = g up to the multiplication of a unimodular constant.

The main goal of this paper is to achieve a parametrization of the solution set of the truncated matricial Hausdorff moment problem in the non-degenerate and degenerate situation. We treat the even and the odd cases simultaneously. Our approach is based on Schur analysis methods. More precisely, we use two interrelated versions of Schur-type algorithms, namely an algebraic one and a function-theoretic one. The algebraic version, worked out in our former paper arXiv:1908.05115, is an algorithm which is applied to finite or infinite sequences of complex matrices. The construction and discussion of the function-theoretic version is a central theme of this paper. This leads us to a complete description via Stieltjes transform of the solution set of the moment problem under consideration. Furthermore, we discuss special solutions in detail.

Using Cauchy's Integral Theorem as a basis, what may be a new series representation for Dirichlet's function η(s) , and hence Riemann's function ζ(s) , is obtained in terms of the Exponential Integral function E s (iκ) of complex argument. From this basis, infinite sums are evaluated, unusual integrals are reduced to known functions and interesting identities are unearthed. The incomplete functions ζ ± (s) and η ± (s) are defined and shown to be intimately related to some of these interesting integrals. An identity relating Euler, Bernouli and Harmonic numbers is developed. It is demonstrated that a known simple integral with complex endpoints can be utilized to evaluate a large number of different integrals, by choosing varying paths between the endpoints.

We prove that every pair of exponential polynomials with imaginary frequencies generates a Poisson-type formula.

We give a complete characterization of closed sets F⊂ R 2 whose distance function d F :=dist(⋅,F) is DC (i.e., is the difference of two convex functions on R 2 ). Using this characterization, a number of properties of such sets is proved.

For parameters c∈(0,1) and β>0 , let ℓ 2 (c,β) be the Hilbert space of real functions defined on N (i.e., real sequences), for which ∥f ∥ 2 c,β := ∑ k=0 ∞ (β ) k k! c k [f(k) ] 2 <∞. We study the best (i.e., the smallest possible) constant γ n (c,β) in the discrete Markov-Bernstein inequality ∥ΔP ∥ c,β ≤ γ n (c,β)∥P ∥ c,β ,P∈ P n , where P n is the set of real algebraic polynomials of degree at most n and Δf(x):=f(x+1)−f(x) . We prove that: (i) γ n (c,1)≤1+ 1 c √ for every n∈N and lim n→∞ γ n (c,1)=1+ 1 c √ . (ii) For every fixed c∈(0,1) , γ n (c,β) is a monotonically decreasing function of β in (0,∞) . (iii) For every fixed c∈(0,1) and β>0 , the best Markov-Bernstein constants γ n (c,β) are bounded uniformly with respect to n . A similar Markov-Bernstein unequality is proved for sequences in ℓ 2 (c,β) . We also establish a relation between the best Markov-Bernstein constants γ n (c,β) and the smallest eigenvalues of certain explicitly given Jacobi matrices.

A family of recently investigated Bernstein functions is revisited and those functions for which the derivatives are logarithmically completely monotonic are identified. This leads to the definition of a class of Bernstein functions, which we propose to call Horn-Bernstein functions because of the results of Roger A. Horn.

We construct a family of Schur multipliers for lower triangular matrices on ℓ p ,1<p<∞ related to θ -summability kernels, a class of kernels including the classical Fejer, Riesz and Bochner kernels. From this simple fact we derive diverse applications. Firstly we find a new class of Schur multipliers for Hankel operators on ℓ 2 , generalizing a result of E. Ricard. Secondly we prove that any space of analytic functions in the unit disc which can be identified with a weighted ℓ p space, has the property that the space of its multipliers is contained in the space of symbols g that induce a bounded generalized Cesáro operator T g .

We provide uniqueness and existence results for the eventually p -convex and eventually p -concave solutions to the difference equation Δf=g on the open half-line (0,∞) , where p is a given nonnegative integer and g is a given function satisfying the asymptotic property that the sequence n↦ Δ p g(n) converges to zero. These solutions, that we call log Γ p -type functions, include various special functions such as the polygamma functions, the logarithm of the Barnes G -function, and the Hurwitz zeta function. Our results generalize to any nonnegative integer p the special case when p=1 obtained by Krull and Webster, who both generalized Bohr-Mollerup-Artin's characterization of the gamma function. We also follow and generalize Webster's approach and provide for log Γ p -type functions analogues of Euler's infinite product, Weierstrass' infinite product, Gauss' limit, Gauss' multiplication formula, Legendre's duplication formula, Euler's constant, Stirling's constant, Stirling's formula, Wallis's product formula, and Raabe's formula for the gamma function. We also introduce and discuss analogues of Binet's function, Burnside's formula, Fontana-Mascheroni's series, Euler's reflection formula, and Gauss' digamma theorem. Lastly, we apply our results to several special functions, including the Hurwitz zeta function and the generalized Stieltjes constants, and show through these examples how powerful is our theory to produce formulas and identities almost systematically.

The Riccati equation method is used to establish a new comparison theorem for systems of two linear first order ordinary differential equation. This result is based on a, so called, concept of "null-classes", and is a generalization of Sturm's comparison theorem.