Wyniki wyszukiwania

Wyszukiwano:
w słowach kluczowych: Riemann-Stieltjes integral

Sortuj według:

Ogranicz wyniki do:

The Basic Existence Theorem of Riemann-Stieltjes Integral

100%

Nakasho K., Narita K., Shidama Y.

Formalized Mathematics

2016

tom 24

nr 4

253-259

In this article, the basic existence theorem of Riemann-Stieltjes integral is formalized. This theorem states that if f is a continuous function and ρ is a function of bounded variation in a closed interval of real line, f is Riemann-Stieltjes integrable with respect to ρ. In the first section, basic properties of real finite sequences are formalized as preliminaries. In the second section, we formalized the existence theorem of the Riemann-Stieltjes integral. These formalizations are based on [15], [12], [10], and [11].

Riemann-Stieltjes Integral

100%

Narita K., Nakasho K., Shidama Y.

Formalized Mathematics

2016

tom 24

nr 3

199-204

In this article, the definitions and basic properties of Riemann-Stieltjes integral are formalized in Mizar [1]. In the first section, we showed the preliminary definition. We proved also some properties of finite sequences of real numbers. In Sec. 2, we defined variation. Using the definition, we also defined bounded variation and total variation, and proved theorems about related properties. In Sec. 3, we defined Riemann-Stieltjes integral. Referring to the way of the article [7], we described the definitions. In the last section, we proved theorems about linearity of Riemann-Stieltjes integral. Because there are two types of linearity in Riemann-Stieltjes integral, we proved linearity in two ways. We showed the proof of theorems based on the description of the article [7]. These formalizations are based on [8], [5], [3], and [4].

Artykuł dostępny w postaci pełnego tekstu - kliknij by otworzyć plik

A sharp companion of Ostrowski’s inequality for the Riemann–Stieltjes integral and applications

100%

Alomari M. W.

Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

2016

tom 15

A sharp companion of Ostrowski’s inequality for the Riemann-Stieltjes integral [...] ∫abf(t) du(t) $\int_a^b {f(t)\;du(t)} $ , where f is assumed to be of r-H-Hölder type on [a, b] and u is of bounded variation on [a, b], is proved. Applications to the approximation problem of the Riemann-Stieltjes integral in terms of Riemann-Stieltjes sums are also pointed out.

Ograniczanie wyników

2 Formalized Mathematics

1 Annales Universitatis Paedagogicae Cracoviensis. Studia Mathematica

2 Nakasho K.

2 Narita K.

2 Shidama Y.

1 Alomari M. W.

3 2016

Wyniki wyszukiwania

The Basic Existence Theorem of Riemann-Stieltjes Integral

Riemann-Stieltjes Integral

A sharp companion of Ostrowski’s inequality for the Riemann–Stieltjes integral and applications