Lebesgue-Stieltjes Integration

International audience The aim of this note is to establish a subclass of $\mathcal{F}$ considered by Segal if functions for which the Ingham-Wintner summability implies $\mathcal{F}$-summability as wide as possible. The subclass is subject to the estimate for the error term of the prime number theorem. We shall make good use of Stieltjes integration which elucidates previous results obtained by Segal.

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Φ V[h] and Riemann-Stieltjes integration

Colloquium Mathematicum ◽

10.4064/cm-60-61-2-421-441 ◽

1990 ◽

Vol 60 (2) ◽

pp. 421-441 ◽

Cited By ~ 5

Author(s):

N. Paul Schembari ◽

Michael Schramm

Keyword(s):

Stieltjes Integration

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Semilinear fractional stochastic differential equations driven by a γ-Hölder continuous signal with γ > 2/3

Stochastics and Dynamics ◽

10.1142/s0219493720500392 ◽

2020 ◽

Vol 21 (01) ◽

pp. 2050039

Author(s):

Jorge A. León ◽

David Márquez-Carreras

Keyword(s):

Acta Math ◽

Stochastic Calculus ◽

Initial Condition ◽

Hölder Continuous ◽

Fractional Stochastic Differential Equations ◽

Stieltjes Integration ◽

Fractal Functions ◽

Fractional Differential ◽

Young Integral ◽

Semilinear Fractional Differential Equation

In this paper, we use the techniques of fractional calculus to study the existence of a unique solution to semilinear fractional differential equation driven by a [Formula: see text]-Hölder continuous function [Formula: see text] with [Formula: see text]. Here, the initial condition is a function that may not be defined at zero and the involved integral with respect to [Formula: see text] is the extension of the Young integral [An inequality of the Hölder type, connected with Stieltjes integration, Acta Math. 67 (1936) 251–282] given by Zähle [Integration with respect to fractal functions and stochastic calculus I, Probab. Theory Related Fields 111 (1998) 333–374].

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