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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.

The Riemann–Stieltjes integral is a generalization of the Riemann integral in which the integrand is integrated with respect to another function, the integrator. The same idea can be used to define the Lebesgue–Stieltjes integral etc.

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How is the Riemann integral a special case of the Stieltjes integral?

From Rudin's Principles of mathematical analysis, 6.2 Definition Let $\alpha$ be a monotonically increasing function on $[a,b]$. ... Corresponding to each partition $P$ of $[a,b]$, we write $$\Delta \alpha_i = \alpha(x_i) - \alpha(x_{i-1}).$$ He…
Tom Stephens
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A very useful lemma for Henstock-Stieltjes integration

I'd like to see a proof (or hints and outlines) for the following lemma, which is very useful to prove some interesting properties, including an Integration by Parts theorem for Henstock-Stieltjes integrals: Let $f$, $g$ and $\varphi$ be (normally…
Pedro Vaz Pimenta
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Almost equivalent definitions of the Riemann–Stieltjes integral

Below, I will present two definitions of the Riemann–Stieltjes integral, the second of which is more general. My question concerns the relationship between these two definitions. Definition 1: Let $f,g:[a,b] \to \mathbb{R}$. For a partition…
MathematicsStudent1122
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Solving Riemann-Stieltjes integral:$\int_{- \pi/4}^{\pi/4} f(x)dg(x)$

I'm having trouble solving this Riemann-Stieltjes integral: $$\int_{- \pi/4}^{\pi/4} f(x)dg(x),$$ where $$f(x):= \begin{cases} \frac{\sin^4x}{\cos^2x}{} &\text{if }x\ge0, \\{}\\ \frac1{\cos^3x} &\text{if }x<0,\end{cases}$$ and $$g(x)=\begin{cases}…
Drake
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Riesz representation theorem for $C([0,1])$

I’m trying to prove the special case of Riesz representation theorem: Every positive (non-negative on non-negative functions) linear continuous functional $\phi$ on the normed space $C([0,1])$ is given by some measure $\mu$ by the…
Ilya
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Let $\alpha$ be an increasing function on $[a,b]$. Show that $\int^a_b\alpha d \alpha = \frac{1}{2}[\alpha (b)^2 - \alpha(a)^2]$

I am wanting to try to prove the question below, but there is a step that I can't get pass. I know that the proof is worthless if I assume incorrectly, and should have stopped proving from there, but I feel that I am close and possibly just missing…
Reuben
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Why is the measure determined by the Stieltjes inversion formula?

$f$ is a Herglotz-Nevanlinna function if it is analytic in the open upper half plane and the imaginary part $\Im f (z) \geq 0$ for $\Im z >0$. $f$ has the following integral representation $$f(z) = a + bz + \int_\mathbb{R} \frac{1}{t-z} -…
user232773
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Riemann-Stieltjes Integral with respect to total variation

In this Inequality for Riemann-Stieltjes integral the following question came up. Suppose functions $f,g:[a,b] \to \mathbb{R}$ are such that $f$ is Riemann-Stieltjes integrable with respect to $g$. Suppose $g$ has bounded variation and $v_a^x(g)$ is…
AlRacoon
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Riemann-Stieltjes integral of unbounded function

In many theorems about the Riemann-Stieltjes integral they required the hypothesis of $f$ to be bounded to then conclude that $f$ is Riemann-Stieltjes integrable. For example, suppose that $f$ is bounded in $I = [a,b]$, $f$ has only finitely many…
user441848
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Do Riemann-Stieltjes integrals "iterate"?

Let's say we define: $$h(x) = \int_a^x f(t)dg(t),$$ then do we have for integrable functions $a$ that: $$\int_a^b a(u) dh(u) = \int_a^b a(u)f(u)dg(u) ?$$ I would like to know whether this holds for either the Riemann-Stieltjes or the…
Chill2Macht
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Hadamard differentiability of function

Let $X$ and $Y$ be Banach spaces. Definition: A function $f:X\rightarrow Y$ is called Hadamard differentiable at $x\in X$ tangentially to $U\subseteq X$ iff $x\in U$ and there exists a continuous linear function $f'_x:U\rightarrow Y$ such that for…
Syd Amerikaner
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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 $$ \frac{1|}{|z-b_1} + \frac{-a_2|}{|z-b_2} + \dots + \frac{-a_n|}{|z-b_n} + \dots,$$ where $a_n,b_n$ are the coefficients of a…
xyz
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Path Signatures and Picard iterations

Recently, I've started studying path signatures and, currently, I'm reading a standard reference, namely "A Primer on the Signature Method in Machine Learning" by Ilya Chevyrev and Andrey Kormilitzin. Now, at some point the authors try to show how…
Oscar
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Relationship between Lebesgue–Stieltjes measure and regular Borel measure

From http://en.wikipedia.org/wiki/Lebesgue%E2%80%93Stieltjes_integration: The Lebesgue–Stieltjes measure is a regular Borel measure, and conversely every regular Borel measure on the real line is of this kind. As far as I know, a…
Du Phan
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Connection between Riemann and Riemann-Stieltjes Integrals

Let functions $f$ and $h$ be Riemann integrable and $H(x) = \int_a^x h(t)dt$. Is it always true that the Riemann-Stieltjes integral $\int_a^bf(x)dH(x)$ exists and $\int_a^b f(x) dH(x) = \int_a^bf(x)h(x) dx$? I remember seeing this used in a…
AlRacoon
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