Estimates of Hellinger integrals of infinitely divisible distributions
Estimates of the remainder in Taylor’s theorem using the Henstock-Kurzweil integral
When a real-valued function of one variable is approximated by its th degree Taylor polynomial, the remainder is estimated using the Alexiewicz and Lebesgue -norms in cases where or are Henstock-Kurzweil integrable. When the only assumption is that is Henstock-Kurzweil integrable then a modified form of the th degree Taylor polynomial is used. When the only assumption is that then the remainder is estimated by applying the Alexiewicz norm to Schwartz distributions of order 1.
Every -integrable function is Pfeffer integrable
Existence theory for integrodifferential equations and Henstock-Kurzweil integral in Banach spaces.
Gauge integrals and series
This note contains a simple example which does clearly indicate the differences in the Henstock-Kurzweil, McShane and strong McShane integrals for Banach space valued functions.
General integration and extensions. I
A general concept of integral is presented in the form given by S. Saks in his famous book Theory of the Integral. A special subclass of integrals is introduced in such a way that the classical integrals (Newton, Riemann, Lebesgue, Perron, Kurzweil-Henstock...) belong to it. A general approach to extensions is presented. The Cauchy and Harnack extensions are introduced for general integrals. The general results give, as a specimen, the Kurzweil-Henstock integration in the form of the extension of...
General integration and extensions.II
This work is a continuation of the paper (Š. Schwabik: General integration and extensions I, Czechoslovak Math. J. 60 (2010), 961–981). Two new general extensions are introduced and studied in the class of general integrals. The new extensions lead to approximate description of the Kurzweil-Henstock integral based on the Lebesgue integral close to the results of S. Nakanishi presented in the paper (S. Nakanishi: A new definition of the Denjoy’s special integral by the method of successive approximation,...
Generalized multiple Perron integrals and the Green-Goursat theorem for differentiable vector fields
Generalized Volterra integral equations
Hake's property of a multidimensional generalized Riemann integral
Henstock-Kurzweil and McShane product integration; descriptive definitions
The Henstock-Kurzweil and McShane product integrals generalize the notion of the Riemann product integral. We study properties of the corresponding indefinite integrals (i.e. product integrals considered as functions of the upper bound of integration). It is shown that the indefinite McShane product integral of a matrix-valued function is absolutely continuous. As a consequence we obtain that the McShane product integral of over exists and is invertible if and only if is Bochner integrable...
Henstock-Kurzweil type integrals in -adic harmonic analysis.
Integrals and Banach spaces for finite order distributions
Let denote the real-valued functions continuous on the extended real line and vanishing at . Let denote the functions that are left continuous, have a right limit at each point and vanish at . Define to be the space of tempered distributions that are the th distributional derivative of a unique function in . Similarly with from . A type of integral is defined on distributions in and . The multipliers are iterated integrals of functions of bounded variation. For each , the spaces...
Integration and decompositions of weak-integrable multifunctions
Conditions guaranteeing Pettis integrability of a Gelfand integrable multifunction and a decomposition theorem for the Henstock-Kurzweil-Gelfand integrable multifunctions are presented.
Integration by Parts
Integrodifferential equations on time scales with Henstock-Kurzweil-Pettis delta integrals.
Kurzweil-Henstock and Kurzweil-Henstock-Pettis integrability of strongly measurable functions
We study the integrability of Banach valued strongly measurable functions defined on . In case of functions given by , where belong to a Banach space and the sets are Lebesgue measurable and pairwise disjoint subsets of , there are well known characterizations for the Bochner and for the Pettis integrability of (cf Musial (1991)). In this paper we give some conditions for the Kurzweil-Henstock and the Kurzweil-Henstock-Pettis integrability of such functions.
Kurzweil-Henstock type integral on zero-dimensional group and some of its applications
A Kurzweil-Henstock type integral on a zero-dimensional abelian group is used to recover by generalized Fourier formulas the coefficients of the series with respect to the characters of such groups, in the compact case, and to obtain an inversion formula for multiplicative integral transforms, in the locally compact case.
Kurzweil’s PU integral as the Lebesgue integral
For a merely continuous partition of unity the PU integral is the Lebesgue integral.
Linear integral equations in the space of regulated functions.