Stieltjes Measure Research Articles

Spectral theory for bounded banded matrices with positive bidiagonal factorization and mixed multiple orthogonal polynomials

Spectral and factorization properties of oscillatory matrices lead to a spectral Favard theorem for bounded banded matrices, that admit a positive bidiagonal factorization, in terms of sequences of mixed multiple orthogonal polynomials with respect to a set positive Lebesgue–Stieltjes measures. A mixed multiple Gauss quadrature formula with corresponding degrees of precision is given.

Local Minimum Principle for an Optimal Control Problem with a Nonregular Mixed Constraint

We consider the simplest optimal control problem with one nonregular mixed constraint $G(x,u)\le0,$ i.e., such a constraint that the gradient $G_u(x, u)$ can vanish on the surface $G = 0.$ Using the Dubovitskii--Milyutin theorem on the approximate separation of convex cones, we prove a first order necessary condition for a weak minimum in the form of the so-called local minimum principle, which is formulated in terms of functions of bounded variation, integrable functions, and Lebesgue--Stieltjes measures and does not use functionals from $(L^\infty)^*$. Two illustrative examples are provided. The work is based on the book by Milyutin [Maximum Principle in the General Problem of Optimal Control, Fizmatlit, Moscow, 2001 (in Russian)].

BV Continuous Solutions of an Evolution Inclusion with Maximal Monotone Operator and Nonconvex-Valued Perturbation. Existence Theorem

An evolution inclusion with the right-hand side containing a time-dependent maximal monotone operator and a multivalued mapping with closed nonconvex values is studied in a separable Hilbert space. The dependence of the maximal monotone operator on time is described with the help of the distance between maximal monotone operators in the sense of Vladimirov. This distance as a function of time has bounded variation with an upper bound given by a nonatomic positive Radon measure. By a solution of the inclusion one means a continuous function of bounded variation whose differential measure (Stieltjes measure) is absolutely continuous with respect to the positive Radon measure above, and the values of the density of this differential measure with respect to the Radon measure belong to the right-hand side of the inclusion almost everywhere. Under the traditional assumptions on the perturbation (measurability, Lipschitzianity in the phase variable in the Hausdorff metric, linear growth condition), the existence of solutions is proven and some properties of the solution set are established.

Subordinators which are infinitely divisible w.r.t. time: Construction, properties, and simulation of max-stable sequences and infinitely divisible laws

The concept of a L\'evy subordinator is generalized to a family of non-decreasing stochastic processes, which are parameterized in terms of two Bernstein functions. Whereas the independent increments property is only maintained in the L\'evy subordinator special case, the considered family is always strongly infinitely divisible with respect to time, meaning that a path can be represented in distribution as a finite sum with arbitrarily many summands of independent and identically distributed paths of another process. Besides distributional properties of the process, we present two applications to the design of accurate and efficient simulation algorithms. First, each member of the considered family corresponds uniquely to an exchangeable max-stable sequence of random variables, and we demonstrate how the associated extreme-value copula can be simulated exactly and efficiently from its Pickands dependence measure. Second, we show how one obtains different series and integral representations for infinitely divisible probability laws by varying the parameterizing pair of Bernstein functions, without changing the law of one-dimensional margins of the process. As a particular example, we present an exact simulation algorithm for compound Poisson distributions from the Bondesson class, for which the generalized inverse of the distribution function of the associated Stieltjes measure can be evaluated accurately.

About the sharpness of the Jensen inequality

The main aim of this paper is to give an improvement of the recent result on the sharpness of the Jensen inequality. The results given here are obtained using different Green functions and considering the case of the real Stieltjes measure, not necessarily positive. Finally, some applications involving various types of f-divergences and Zipf–Mandelbrot law are presented.

On the Relationship between the Dimension of the Lebesgue–Stieltjes Measure and the Rate of Approximation of a Function by Step Functions

The relationship between the rate of approximation of a monotone function by step functions (with an increasing number of values) and the Hausdorff dimension of the corresponding Lebesgue–Stieltjes measure is studied. An upper bound on the dimension is found in terms of the approximation rate, and it is shown that a lower bound cannot be constructed in these terms.

Combining Continuous and Discrete Phenomena for Feynman’s Operational Calculus in the Presence of a ( $${\varvec{C}}_{\mathbf{0 }}$$ C 0 ) Semigroup and Feynman–Kac Formulas with Lebesgue–Stieltjes Measures

This paper introduces the presence of a $$(C_{0})$$ semigroup of linear operators into the disentangling of functions of noncommuting operators in the setting where time-ordering measures with finitely supported discrete parts are allowed. Some examples are discussed. Furthermore, in this more general setting the main result of this paper, an evolution equation satisfied by disentangled exponential functions is obtained. A related generalized integral equation is also discussed. Via some detailed examples, Feynman–Kac formulas (Volterra–Stieltjes integral equations) with Lebesgue–Stieltjes measures are also derived and an associated differential equation is derived.

Imprecise Vector: The Case of Subnormality

Imprecise vector is a vector containing imprecise elements. An array or a vector $$X=(X_{1}, X_{2},\ldots,X_{n})$$ is an imprecise vector if the elements $$X_{i}, i=1,2,\ldots,n,$$ are imprecise numbers. Two laws of randomness are necessary and sufficient to define a normal law of impreciseness. Based on the method of superimposition of sets, the construction of the membership surface of normal imprecise vector has been developed with reference to probability measure. A normal imprecise vector is a special case of a subnormal imprecise vector in the sense that a subnormal imprecise vector is nothing but a generalized imprecise vector. The method of construction of the membership surface of subnormal imprecise vector with reference to Lebesgue–Stieltjes measure is explained in this paper.

New Harnack inequalities and existence theorems for radially symmetric solutions of elliptic PDEs with sign changing or vanishing Green's function

We consider the perturbed Hammerstein integral equationy(t)=γ1(t)H1(φ1(y))+γ2(t)H2(φ2(y))+λ∫01G(t,s)f(s,y(s))ds, where φ1 and φ2 are linear functionals realized as Stieltjes integrals with associated signed Stieltjes measures. We demonstrate that by utilizing a nonstandard order cone together with an associated nonstandard open set, existence of a positive solution to the integral equation can be guaranteed under relatively mild hypotheses on the various constituent functions and functionals in the case in which the Green's function (t,s)↦G(t,s) is allowed to both vanish and change sign. As an example we apply our results to radially symmetric elliptic PDEs of the form−Δu=λh(|x|)g(u(x)), |x|∈[R1,R2]∂∂ru(x)|x∈∂BR2=0(u(R1x)+u(Rηx))|x∈∂B1=H(φ(u)), where 0<R1<Rη<R2<+∞, φ is a linear functional, and H is a continuous function. As part of this analysis we demonstrate a new methodology for constructing a suitable Harnack-like inequality associated to the boundary value problem.

Pisot numbers and strong negations

We introduce and study representation systems for the numbers in the unit interval [0, 1]. We call them ϕm-systems (where ϕm is a pseudo-golden ratio). With the aid of these representation systems, we define a family hm of strong negations and an increasing function gm which is the inverse of the generator of hm. The functions hm and gm are singular, and we study several properties; among which we calculate the Hausdorff dimensions of certain sets that are related to them. Finally, we prove that gm is an infinite convolution, and the sequence of coefficients in the Fourier series of its associated Stieltjes measure does not converge to zero.

Levinson\u2019s type generalization of the Jensen inequality and its converse for real Stieltjes measure

We derive the Levinson type generalization of the Jensen and the converse Jensen inequality for real Stieltjes measure, not necessarily positive. As a consequence, also the Levinson type generalization of the Hermite-Hadamard inequality is obtained. Similarly, we derive the Levinson type generalization of Giaccardi’s inequality. The obtained results are then applied for establishing new mean-value theorems. The results from this paper represent a generalization of several recent results.

Approximation and its implementation process of the stochastic hybrid fuzzy system

The stochastic Mamdani and Takagi–Sugeno fuzzy systems are firstly unified in a random environment, and the resulting stochastic hybrid fuzzy system is established according to some stochastic parameters. Secondly, A canonical representation of the stochastic process with orthogonal increments is presented by the properties of the Lebesgue–Stieltjes measure and stochastic integral, the approximation of the stochastic hybrid fuzzy system in the mean square sense is proved. Finally, an implementation process of this system is described through a simulation example, and the surface figure of the covariance function shows that the stochastic hybrid fuzzy system has excellent approximation capability.

Note on the differentiability of singular Lebesgue–Stieltjes measures

Abstract For an arbitrary translation invariant basis B which does not differentiate an integral of some function, we construct a singular Lebesgue–Stieltjes measure having non-vanishing means with respect to B at almost every point.

E2-quasi-exact solvability for non-Hermitian models

We propose the notion of E2-quasi-exact solvability and apply this idea to find explicit solutions to the eigenvalue problem for a non-Hermitian Hamiltonian system depending on two parameters. The model considered reduces to the complex Mathieu Hamiltonian in a double scaling limit, which enables us to compute the exceptional points in the energy spectrum of the latter as a limiting process of the zeros for some algebraic equations. The coefficient functions in the quasi-exact eigenfunctions are univariate polynomials in the energy obeying a three-term recurrence relation. The latter property guarantees the existence of a linear functional such that the polynomials become orthogonal. The polynomials are shown to factorize for all levels above the quantization condition leading to vanishing norms rendering them to be weakly orthogonal. In two concrete examples we compute the explicit expressions for the Stieltjes measure.

A new non-Hermitian E2-quasi-exactly solvable model

We construct a previously unknown $E_2$-quasi-exactly solvable non-Hermitian model whose eigenfunctions involve weakly orthogonal polynomials obeying three-term recurrence relations that factorize beyond the quantization level. The model becomes Hermitian when one of its two parameters is fixed to a specific value. We analyze the double scaling limit of this model leading to the complex Mathieu equation. The norms, Stieltjes measures and moment functionals are evaluated for some concrete values of one of the two parameters.

A note on semipositone boundary value problems with nonlocal, nonlinear boundary conditions

We consider the existence of at least one positive solution to a semipositone boundary value problem with nonlocal, nonlinear boundary conditions, which can be quite general since the nonlinearity is realized as a Stieltjes integral. By assuming that the associated Stieltjes measure decomposes in a certain way, the classical Leray-Schauder degree is utilized to derive the existence result.

Characterizations of monotone [formula omitted]-regularly varying functions by means of indefinite eigenvalue problems and HELP type inequalities

Connections between different areas of mathematical analysis are obtained: regular variation, indefinite operator theory and HELP type inequalities. Using a result by Parfyonov, the nondecreasing and so-called positively increasing functions induce precisely those indefinite Kreĭn–Feller eigenvalue problems such that the eigenfunctions have the Riesz basis property. Furthermore, these functions induce precisely those Lebesgue–Stieltjes measures such that the associated HELP type inequalities are valid. The so-called dual Kreĭn–Feller eigenvalue problems allow a similar characterization of the class of nondecreasing O-regularly varying functions.

Moments of discrete measures with dense jumps induced by [formula omitted]-expansions

Let β>1. Through an appeal to β-expansions we define a strictly increasing and left-continuous function μβ on [0,1]. Then μβ turns out to be a pure jump distribution. In other words, its associated Lebesgue–Stieltjes measure is discrete, i.e., a summation of point masses. The present note studies the moment of this discrete measure and its asymptotics.

Non-local PDEs with a state-dependent delay term presented by Stieltjes integral

Parabolic partial differential equations with state-dependent delays (SDDs) are investigated. The delay term presented by Stieltjes integral simultaneously includes discrete and distributed SDDs. The singular Lebesgue–Stieltjes measure is also admissible. The conditions for the corresponding initial value problem to be well-posed are presented. The existence of a compact global attractor is proved.

Quantum field theory, gravity and cosmology in a fractal universe

We propose a model for a power-counting renormalizable field theory living in a fractal spacetime. The action is Lorentz covariant and equipped with a Stieltjes measure. The system flows, even in a classical sense, from an ultraviolet regime where spacetime has Hausdorff dimension 2 to an infrared limit coinciding with a standard $D$-dimensional field theory. We discuss the properties of a scalar field model at classical and quantum level. Classically, the field lives on a fractal which exchanges energy-momentum with the bulk of integer topological dimension D. Although an observer experiences dissipation, the total energy-momentum is conserved. The field spectrum is a continuum of massive modes. The gravitational sector and Einstein equations are discussed in detail, also on cosmological backgrounds. We find ultraviolet cosmological solutions and comment on their implications for the early universe.