Search results for "Henstock Integral"
showing 10 items of 33 documents
Variational Henstock integrability of Banach space valued functions
2016
We study the integrability of Banach space valued strongly measurable functions defined on $[0,1]$. In the case of functions $f$ given by $\sum \nolimits _{n=1}^{\infty } x_n\chi _{E_n}$, where $x_n $ are points of a Banach space and the sets $E_n$ are Lebesgue measurable and pairwise disjoint subsets of $[0,1]$, there are well known characterizations for Bochner and Pettis integrability of $f$. The function $f$ is Bochner integrable if and only if the series $\sum \nolimits _{n=1}^{\infty }x_n|E_n|$ is absolutely convergent. Unconditional convergence of the series is equivalent to Pettis integrability of $f$. In this paper we give some conditions for variational Henstock integrability of a…
Set valued Kurzweil-Henstock-Pettis integral
2005
It is shown that the obvious generalization of the Pettis integral of a multifunction obtained by replacing the Lebesgue integrability of the support functions by the Kurzweil--Henstock integrability, produces an integral which can be described -- in case of multifunctions with (weakly) compact convex values -- in terms of the Pettis set-valued integral.
Differentiation of an additive interval measure with values in a conjugate Banach space
2014
We present a complete characterization of finitely additive interval measures with values in conjugate Banach spaces which can be represented as Henstock-Kurzweil-Gelfand integrals. If the range space has the weak Radon-Nikodým property (WRNP), then we precisely describe when these integrals are in fact Henstock-Kurzweil-Pettis integrals.
Multidimensional dyadic Kurzweil–Henstock- and Perron-type integrals in the theory of Haar and Walsh series
2015
Abstract The problem of recovering the coefficients of rectangular convergent multiple Haar and Walsh series from their sums, by generalized Fourier formulas, is reduced to the one of recovering a function (the primitive) from its derivative with respect to the appropriate derivation basis. Multidimensional dyadic Kurzweil–Henstock- and Perron-type integrals are compared and it is shown that a Perron-type integral, defined by major and minor functions having a special continuity property, solves the coefficients problem for series which are convergent everywhere outside some uniqueness sets.
Kurzweil-Henstock type integration on Banach spaces
2004
In this paper properties of Kurzweil-Henstock and Kurzweil-Henstock-Pettis integrals for vector-valued functions are studied. In particular, the absolute integrability for Kurzweil-Henstock integrable functions is characterized and a Kurzweil-Henstock version of the Vitali Theorem for Pettis integrable functions is given.
Strongly measurable Kurzweil-Henstock type integrable functions and series
2008
We give necessary and sufficient conditions for the scalar Kurzweil-Henstock integrability and the Kurzweil-Henstock-Pettis integrability of functions $f:[1, infty) ightarrow X$ defined as $f=sum_{n=1}^infty x_n chi_{[n,n+1)}$. Also the variational Henstock integrability is considered
A Hake-Type Theorem for Integrals with Respect to Abstract Derivation Bases in the Riesz Space Setting
2015
Abstract A Kurzweil-Henstock type integral with respect to an abstract derivation basis in a topological measure space, for Riesz space-valued functions, is studied. A Hake-type theorem is proved for this integral, by using technical properties of Riesz spaces.
A variational henstock integral characterization of the radon-nikodým property
2009
A characterization of Banach spaces possessing the Radon-Nikodym property is given in terms of finitely additive interval functions. We prove that a Banach space X has the RNP if and only if each X-valued finitely additive interval function possessing absolutely continuous variational measure is a variational Henstock integral of an X-valued function. Due to that characterization several X-valued set functions that are only finitely additive can be represented as integrals.
Approximation of Banach space valued Riemann type integrable functions by step functions
2008
In this talk we consider the possibility to approximate (with respect to some topology induced by the Alexiewicz norm) non absolutely integrable functions defined on the unit interval by step functions. In particular we show that any Henstock (respectively Henstock-Kurzweil-Pettis, Denjoy-Khintchine-Pettis) integrable functions can be scalarly approximate in the Alexiewicz norm by a sequence of step functions. Moreover the approximation may be done in the Alexiewicz norm if and only if the range of the integral is relatively norm compact (property which is automatically satisfied by the Henstock integrable functions). We also provide an example to show that, unlike the Pettis case, Henstock…
A decomposition of Denjoy-Khintchine-Pettis and Henstock-Kurzweil-Pettis integrable multifunctions
2010
We proved in one of our earlier papers that in case of separable Banach space valued multifunctions each Henstock-Kurzweil-Pettis integrable multifunction can be represented as a sum of one of its Henstock-Kurzweil-Pettis integrable selector and a Pettis integrable multifunction. Now, we prove that the same result can be achieved in case of an arbitrary Banach space. Moreover we show that an analogous result holds true also for the Denjoy-Khintchine-Pettis integrable multifunctions. Applying the representation theorem we describe the multipliers of HKP and DKP integrable functions. Then we use this description to obtain an operator characterization of HKP and DKP integrability.