# Questions tagged [stieltjes-integral]

For questions about Stieltjes integrals. Use with other tags as needed, such as [riemann-integration], to specify Riemann–Stieltjes, Lebesgue–Stieltjes, etc.

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### Riemann-Stieltjes Integral Does not Add Up

Does there exist functions $f,\phi : [a,b]\rightarrow \mathbb{R}$ and $c\in [a,b]$ such that the Riemann-Stieltjes integrals $\int_a^c f\,d\phi$ and $\int_c^b f\,d\phi$ exist but $\int_a^b f\,d\phi$ ...
1 vote
23 views

### Is there a modification of Laplace's method for obtaining the asymptotic behaviour of Riemann-Stieltjes integrals?

Is there a modification of Laplace's method for obtaining the asymptotic behaviour of Riemann-Stieltjes integrals? In particular, I am interested in asymptotic behaviour of the Riemann-Stieltjes ...
1 vote
64 views

### A Lebesgue Stieltjes integral where the integrator is a Lebesgue Stieltjes integral

Let $f: \mathbb{R} \to \mathbb{R}$ be a Borel measurable and non-negative function; let $g: \mathbb{R} \to \mathbb{R}$ have bounded variation but not necessarily be right-continuous. Then, the ...
322 views

### Proving that quotient of orthogonal polynomials is a Padé approximant of Stieltjes transform.

Conext and notation. In the question below, $R_n(z) / S_n(z)$ denotes the n-th convergent of the continued fraction $$\frac{1|}{|z-b_1} + \frac{-a_2|}{|z-b_2} + \dots + \frac{-a_n|}{|z-b_n} + \dots,$$...
1 vote
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### Total variation of a class in $L^1 [0, 1]$ and lower semicontinuity

Consider a function $f \in \mathcal L^1 [0, 1]$: we define the total variation of $f$ as usual by $$V_0^1 (f) = \sup \sum_{k = 0}^{n - 1} | f(x_{k + 1}) - f(x_k) |,$$ where the supremum is taken ...
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### Completely Monotonic Functions and Stieltjes-like integral

A function $f(x)$ is completely monotonic if $(-1)^nf^{(n)}(x) \geq 0$ for all $n \geq 0$ and all $x > 0$. These functions are characterized by Bernstein's theorem and are well studied in Widder's ...
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### question about definition of Lebesgue-Stieltjes measure

In some books it's defined with the outer measure: $$\mu^*(E)=\inf\{\sum (f(b_n)-f(a_n)):E\subset\bigcup (a_n, b_n]\}$$ where $f: \mathbb{R}\to\mathbb{R}$ is not-decreasing and right continuous. I'm ...
487 views

### If the Riemann-Stieltjes integral $\int_a^b{f\,dg}$ exists for every $f\in\mathcal{R}[a,b]$, then $g$ is absolutely continuous

In Remark 18.13 of Real Analysis: Foundations and Functions of One Variable, the author states that The class of Riemann integrable functions and the class of absolutely continuous functions are also ...
1 vote
153 views

### A question on integrating a function, with respect to another function?

I've been asked to find the area described by $$A={ (x,y), y^2 \leqslant2x; and, y\geqslant 4x-1 }$$ which is effectively the following area(If I can insert a graph using mathJax, please let me ...
1 vote
106 views

### How does this theorem on continued fractions relate to the Riemann-Stieltjes integral?

I'm going to include two precursor sections here to introduce all of the content referenced in my actual question (if you're familiar with both continued fractions and the Riemann-Stieltjes integral, ...
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### Total variation of Lebesgue-Stieltjes integral

Setting We work on a filtered probability space. Let $A$ be a process of finite variation (FV) started from $0$ and $H$ be predictable, positive and such that $H\cdot A$ is defined $\omega$-wise as a ... 1 vote
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### Lebesgue–Stieltjes integration with discontinuous function

My question comes from the book Credit risk modeling: Valuation and hedging by Tomasz R. Bielecki and Marek Rutkowski, Page 137 at the bottom. It is easily described as follows: Consider a probability ...
1 vote
39 views

### How do I integrate a not necessarily differentiable multivariate (Riemann)-Stieltjes integral?

I am having a bit of trouble grasping the way of calculating a Stieltjes integral where the differential term might not be at least once differentiable with respect to every variable. I have a good ...
94 views

### Regarding Theorem 1.9, Conway, complex Analysis

I have a doubt in equation (1.11) in Chapter 4, Section 1 of J.B Conway’s, functions of one complex variable. I know that the estimation of the integral in equation 1.10, comes from the previous ...
1 vote
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### Convert probability into an integral with respect to a distribution function

I'm studying a proof and have problems with a step where a probability is converted into a Lebesgue Stieltjes integral with respect to a distribution function. Let $(X_i)$ be a sequence of independent,...
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### Resources for Riemann-Stieltjes or Lebesgue-Stieltjes integrals

I'm an engineer, who works with retarded delay differential equations. So far I've encountered systems with single or multiple point-wise delays, and systems with distributed delays. Someone on a ...
1 vote
In the book of Jean Prüss (Evolutionary Integral Equations and Applications, 2012, pg. 111) it says in the Proposition 4.10... Let $dc$ be completely positive, let $w(t;\tau)$ denote the associated ...