Related papers: Numerous approximations of Riemann-Stieltjes doubl…
Finite-part integration is a recently introduced method of evaluating convergent integrals by means of the finite part of divergent integrals [E.A. Galapon, {\it Proc. R. Soc. A 473, 20160567} (2017)]. Current application of the method…
We write expressions connected with numerical differentiation formulas of order $2$ in the form of Stieltjes integral, then we use Ohlin lemma and Levin-Stechkin theorem to study inequalities connected with these expressions. In particular,…
In this work, we extend the concept of the Stieltjes derivative to encompass left-continuous derivators with bounded variation, thereby relaxing the monotonicity constraint. This generalization necessitates a refined definition of the…
Various kinds of Stieltjes integrals using gauge integration have become highly popular in the field of differential equations and other applications. In the theories of integration and of ordinary differential equations, convergence…
We define integrals for functions on finite-dimensional algebras, adapting methods from Leinster's research. This paper discusses the relationships between the integrals of functions defined on subsets $\mathbb{I}_1 \subseteq…
In this paper we provided a classification for partitions of intervals on the hyperbolic plane. Given a partition, to be named strong, we define a notion of a hyperbolic-valued functions of bounded variation and a kind of Riemann-Stieltjes…
The paper has two relatively distinct but connected goals; the first is to define the notion of Pad\'e\ approximation of Weyl-Stiltjes transforms on an arbitrary compact Riemann surface of higher genus. The data consists of a contour in the…
In this paper, new sharp weighted generalizations of Ostrowski and generalized trapezoid type inequalities for the Riemann--Stieltjes integrals are proved. Several related inequalities are deduced and investigated. New Simpson's type…
We offer some comments on series involving the M$\ddot{o}$bius function which approximate sums over primes. To accomplish this, we utilize the derivative of the Gram series by applying Riemann-Stieltjes integration. We offer a new formula…
In the theory of time scales, given $\mathbb{T}$ a time scale with at least two distinct elements, an integration theory is developed using ideas already well known as Riemann sums. Another, more daring, approach is to treat an integration…
It is known in the case of the Stieltjes transform that evaluating the integral by expanding the kernel of transformation followed by term by term integration leads to an infinite series of divergent integrals. Moreover, it is known that…
We establish a new perturbation theory for orthogonal polynomials using a Riemann--Hilbert approach and consider applications in numerical linear algebra and random matrix theory. This new approach shows that the orthogonal polynomials with…
This essay explores the meaning of stochastic differential equations and stochastic integrals. It sets these subjects in a context of Riemann-Stieltjes integration. It is intended as a comment or supplement to \cite{MTRV}.
In this paper we solve the problem of approximating functionals $(\varphi(A)x, f)$ (where $\varphi(A)$ is some function of self-adjoint operator $A$) on the class of elements of a Hilbert space that is defined with the help of another…
The integration operators (*) $({\mathcal J}^+\,g)(x) = \int_a^x g(t) \, dt$ and (**) $({\mathcal J}^-\,g)(x) = \int_x^b g(t) \, dt$ defined on an interval $(a,b) \subseteq {\mathbf R}$ yield new identities for indefinite convolutions,…
We consider general formulations of the change of variable formula for the Riemann-Stieltjes integral, including the case when the substitution is not invertible.
We introduce two novel numerical approaches for computing Feynman integrals based on their complete monotonicity (CM) and Stieltjes properties. The first method uses that scalar Feynman integrals are CM, meaning that all their derivatives…
By using some basic calculus of multiple integration, we provide an alternative expression of the integral $$ \int_{]a,b[^n} f(\mathbf{x},\min x_i,\max x_i) d\mathbf{x}, $$ in which the minimum and the maximum are replaced with two single…
The Gauss-Bonnet Formula is a significant achievement in 19th century differential geometry for the case of surfaces and the 20th century cumulative work of H. Hopf, W. Fenchel, C. B. Allendoerfer, A. Weil and S.S. Chern for…
The paper surveys the basic properties of generalized Stieltjes functions including some new ones. We introduce the notion of the exact Stieltjes order and give a criterion of exactness, simple sufficient conditions and some prototypical…