the Riemann–Stieltjes integral is a generalization of the Riemann integral, named after Bernhard Riemann and Thomas Joannes Stieltjes. The definition...
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framework. The Lebesgue–Stieltjes integral is the ordinary Lebesgue integral with respect to a measure known as the Lebesgue–Stieltjes measure, which may be...
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Thomas Stieltjes Institute for Mathematics at Leiden University, dissolved in 2011, was named after him, as is the Riemann–Stieltjes integral. Stieltjes was...
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Henstock–Kurzweil integral or generalized Riemann integral or gauge integral – also known as the (narrow) Denjoy integral (pronounced [dɑ̃ʒwa]), Luzin integral or Perron...
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Lebesgue–Stieltjes integral, further developed by Johann Radon, which generalizes both the Riemann–Stieltjes and Lebesgue integrals. The Daniell integral, which...
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Borel measure (section Lebesgue–Stieltjes integral)
products. The Lebesgue–Stieltjes integral is the ordinary Lebesgue integral with respect to a measure known as the Lebesgue–Stieltjes measure, which may be...
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Itô calculus (redirect from Itô integral)
central concept is the Itô stochastic integral, a stochastic generalization of the Riemann–Stieltjes integral in analysis. The integrands and the integrators...
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The Laplace–Stieltjes transform, named for Pierre-Simon Laplace and Thomas Joannes Stieltjes, is an integral transform similar to the Laplace transform...
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Riemann–Stieltjes integral, and most disappear with the Lebesgue integral, though the latter does not have a satisfactory treatment of improper integrals. The...
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Fourier transform (redirect from Fourier integral)
\xi }\,d\mu ,} and called the Fourier-Stieltjes transform due to its connection with the Riemann-Stieltjes integral representation of (Radon) measures....
177 KB (21,313 words) - 19:14, 8 July 2025
Riemann–Stieltjes integral). The circle ∘ {\displaystyle \circ } is a notational device, used to distinguish this integral from the Itô integral. Many integration...
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f^{-1}} is not differentiable: it suffices, for example, to use the Stieltjes integral in the previous argument. On the other hand, even though general monotonic...
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: 13–15 Other integrals can be approximated by versions of the Gaussian integral. Fourier integrals are also considered. The first integral, with broad...
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Integration by parts (category Integral calculus)
formulations of integration by parts exist for the Riemann–Stieltjes and Lebesgue–Stieltjes integrals. The discrete analogue for sequences is called summation...
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Length Area Volume Probability Moving average Riemann sum Riemann–Stieltjes integral Bounded variation Jordan content Cauchy principal value Measure (mathematics)...
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Riemann–Stieltjes integral, along with an appropriate function of bounded variation, gives a definition of integral equivalent to the Lebesgue–Stieltjes integral...
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It was conjectured by Thomas Joannes Stieltjes, in an 1885 letter to Charles Hermite (reprinted in Stieltjes (1905)), and again in print by Franz Mertens (1897)...
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integrator in a Stieltjes integral is represented as the differential of a function. Formally, the differential appearing under the integral behaves exactly...
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Kolmogorov integral (or Kolmogoroff integral) is a generalized integral introduced by Kolmogoroff (1930) including the Lebesgue–Stieltjes integral, the Burkill...
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bounded, continuous function g: R → R, where the integrals involved are Riemann–Stieltjes integrals. Note that if X and X1, X2, ... are random variables...
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Measure Sigma-algebra Lebesgue space Lebesgue–Stieltjes integration Riemann integral Henstock–Kurzweil integral This approach can be found in most treatments...
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integral in his habilitation. Among other things, he showed that every piecewise continuous function is integrable. Similarly, the Stieltjes integral...
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variation are precisely those with respect to which one may find Riemann–Stieltjes integrals of all continuous functions. Another characterization states that...
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the Darboux integral, rather than the true Riemann integral. Moreover, the definition is readily extended to defining Riemann–Stieltjes integration....
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constant. We can use a combination of a Möbius transformation and the Stieltjes inversion formula to construct the holomorphic function from the real...
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Laplace transform (category Integral transforms)
}{2}}.} The (unilateral) Laplace–Stieltjes transform of a function g : ℝ → ℝ is defined by the Lebesgue–Stieltjes integral { L ∗ g } ( s ) = ∫ 0 ∞ e − s...
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Dirac delta function (section Indefinite integral)
against a continuous function can be properly understood as a Riemann–Stieltjes integral: ∫ − ∞ ∞ f ( x ) δ ( d x ) = ∫ − ∞ ∞ f ( x ) d H ( x ) . {\displaystyle...
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FX of X, with the expected value of g(X) now given by the Lebesgue–Stieltjes integral E [ g ( X ) ] = ∫ − ∞ ∞ g ( x ) d F X ( x ) . {\displaystyle \operatorname...
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Partitions are used in the theory of the Riemann integral, the Riemann–Stieltjes integral and the regulated integral. Specifically, as finer partitions of a given...
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of bounded variation. The Laplace–Stieltjes transform of F {\displaystyle F} is defined by the Stieltjes integral ω ( s ) = ∫ 0 ∞ e − s t d F ( t ) ...
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