A Riemann-type theorem for a Riemann-type integral

For functions which are Henstock–Kurzweil integrable but not Lebesgue-integrable we prove a theorem which resembles the Riemann theorem on the rearrangement of conditionally convergent series.

     

The McShane, PU and Henstock integrals of Banach valued functions

Some relationships between the vector valued Henstock and McShane integrals are investigated. An integral for vector valued functions, defined by means of partitions of the unity (the PU-integral) is studied. In particular it is shown that a vector valued function is McShane integrable if and only if it is both Pettis and PU-integrable. Convergence theorems for the Henstock variational and the PU integrals are stated. The families of multipliers for the Henstock and the Henstock variational integrals of vector valued functions are characterized.     

关于Banach-值函数的Henstock积分

讨论了Banach-值函数的H enstock积分与P ettis积分的关系,证明了若f是几乎可分值且弱有界的,则f是H enstock可积的当且仅当f是P ettis可积的.     

Decomposability in the space of HKP‐integrable functions

In this paper we introduce the notion of decomposability in the space of Henstock‐Kurzweil‐Pettis integrable (for short HKP‐integrable) functions. We show representations theorems for decomposable sets of HKP‐integrable or Henstock integrable functions, in terms of the family of selections of suitable multifunctions.     

Relations among Henstock,McShane and Pettis integrals for multifunctions with compact convex values

Fremlin (Ill J Math 38:471–479, 1994) proved that a Banach space valued function is McShane integrable if and only if it is Henstock and Pettis integrable. In this paper we prove that the result remains valid also in case of multifunctions with compact convex values being subsets of an arbitrary Banach space (see Theorem 3.4). Di Piazza and Musia? (Monatsh Math 148:119–126, 2006) proved that if $X$ is a separable Banach space, then each Henstock integrable multifunction which takes as its values convex compact subsets of $X$ is a sum of a McShane integrable multifunction and a Henstock integrable function. Here we show that such a decomposition is true also in case of an arbitrary Banach space (see Theorem 3.3). We prove also that Henstock and McShane integrable multifunctions possess Henstock and McShane (respectively) integrable selections (see Theorem 3.1).     

Properties of functions generalized convex with respect to a WT-system

Continuity and convergence properties of functions, generalized convex with respect to a continuous weak Tchebysheff system, are investigated. It is shown that, under certain non-degeneracy assumptions on the weak Tchebysheff system, every function in its generalized convex cone is continuous, and pointwise convergent sequences of generalized convex functions are uniformly convergent on compact subsets of the domain. Further, it is proved that, with respect to a continuous Tchebysheff system, L_p-convergence to a continuous function, pointwise convergence and uniform convergence of a sequence of generalized convex functions are equivalent on compact subsets of the domain.     

GENERALIZED RIEMANN INTEGRAL ON FRACTAL SETS

The theory of integration to mathematical analysis is so important that many mathematicians continue to develop new theory to enlarge the class of integrable functions and simplify the Lebesgue theory integration. In this paper, by slight modifying the definition of the Henstock integral which was introduced by Jaroslav Kurzweil and Ralph Henstock, we present a new definition of integral on fractal sets. Furthermore, its integrability has been discussed, and the relationship between differentiation and integral is also established. As an example, the integral of Cantor function on Cantor set is calculated.     

绝对Henstock可积函数都是Mcshane可积的

我们给出每个绝对Ｈｅｎｓｔｏｃｋ可积函数都是Ｍｃｓｈａｎｅ可积的一个新的证明     

Some new results on integration for multifunction

It has been proven in Di Piazza and Musia? (Set Valued Anal 13:167–179, 2005, Vector measures, integration and related topics, Birkhauser Verlag, Basel, vol 201, pp 171–182, 2010) that each Henstock–Kurzweil–Pettis integrable multifunction with weakly compact values can be represented as a sum of one of its selections and a Pettis integrable multifunction. We prove here that if the initial multifunction is also Bochner measurable and has absolutely continuous variational measure of its integral, then it is a sum of a strongly measurable selection and of a variationally Henstock integrable multifunction that is also Birkhoff integrable (Theorem 3.4). Moreover, in case of strongly measurable (multi)functions, a characterization of the Birkhoff integrability is given using a kind of Birkhoff strong property.     

Henstock积分问题的一个注记

本文证明了如果X是不含c0的Banach空间，f是定义在区间I0包含R^m上取值于Panach空间X的函数，并且，在I0上Henstock可积，则总存在I0的一个非退化子区间J，使得f在J上McShane可积，从而对Kartak的一个问题作出了肯定的回答．     

THE HENSTOCK INTEGRAL FOR FUNCTIONS WITH VALUES IN NUCLEAR SPACES AND THE HENSTOCK LEMMA

This paper shows that Henstock‘s Lemma holds for functions with values in a countably Hilbert space, where the Henstock integral is defined as a natural extension of the resl valued case.     

A connection between a general class of superparabolic functions and supersolutions

We show for a general class of parabolic equations that every bounded superparabolic function is a weak supersolution and, in particular, has derivatives in a Sobolev sense. To this end, we establish various comparison principles between super- and subparabolic functions, and show that a pointwise limit of uniformly bounded weak supersolutions is a weak supersolution.     

A Decomposition Theorem for Compact-Valued Henstock Integral

We prove that if X is a separable Banach space, then a measurable multifunction Γ : [0, 1] → ck(X) is Henstock integrable if and only if Γ can be represented as Γ = G + f, where G : [0, 1] → ck(X) is McShane integrable and f is a Henstock integrable selection of Γ.     

UNIFORM APPROXIMATION OF A MULTIOBJECTIVE OPTIMIZATION SEQUENCE

In this paper the Pareto efficiency of a uniformly convergent multiobjective optimization sequence is studied. We obtain some relation between the Pareto efficient solutions of a given multiobjective optimization problem and those of its uniformly convergent optimization sequence and also some relation between the weak Pareto efficient solutions of the same optimization problem and those of its uniformly convergent optimization sequence. Besides, under a compact convex assumption for constraints set and a certain convex assumption for both objective and constraint functions, we also get some sufficient and necessary conditions that the limit of solutions of a uniformly convergent multiobjective optimization sequence is the solution of a given multiobjective optimization problem.     

POINTWISE CONVERGENCE FOR EXPANSIONS IN SPHERICAL MONOGENICS

We offer a new approach to deal with the pointwise convergence of FourierLaplace series on the unit sphere of even-dimensional Euclidean spaces. By using spherical monogenics defined through the generalized Cauchy-Riemann operator, we obtain the spherical monogenic expansions of square integrable functions on the unit sphere. Based on the generalization of Fueter＇s theorem inducing monogenic functions from holomorphic functions in the complex plane and the classical Carleson＇s theorem, a pointwise convergence theorem on the new expansion is proved. The result is a generalization of Carleson＇s theorem to the higher dimensional Euclidean spaces. The approach is simpler than those by using special functions, which may have the advantage to induce the singular integral approach for pointwise convergence problems on the spheres.     

Compactness theory and mappings with finite length distortion

The present paper is devoted to the study of mappings with finite length distortion introduced in 2004 by O. Martio, V. Ryazanov, U. Srebro, and E. Yakubov. It is proved that the locally uniform limit of homeomorphisms with finite length distortion is a homeomorphism or a constant provided that the so-called inner dilatations of the sequence of homeomorphisms are almost everywhere (a.e.) majorized by a locally integrable function. In particular, it is studied the pointwise behavior of the so-called outer dilatations. For these dilatations, the pointwise semicontinuity and semicontinuty in the mean are proved. It is also proved some theorems on the convergence of matrix dilatations.     

Band-limited signal extrapolation by truncated Bernstein polynomials

An algorithm is given for everywhere extrapolating a band-limited signal known only on an interval of arbitrary finite length. The scheme utilizes a finite number of equally spaced samples of the given function and provides a time-limited polynomial approximation. The approximation functions are shown to converge everywhere pointwise and uniformly in any compact interval to the band-limited signal. When the original band-limited signal is also Lebesgue integrable it is also established that the Fourier transform of the approximating signal converges uniformly to the Fourier transform of the original signal.     

Fatou's identity and Lebesgue's convergence theorem

The classical Fatou lemma for bounded sequences of nonnegative integrable functions is represented as an equality. A similar result is stated for measure convergent sequences. Neither result requires a uniform integrability assumption. For the latter a converse is proven. Two extensions of Lebesgue's convergence theorem are presented.

     

Integration of Banach-valued functions and Haar series with Banach-valued coefficients

It is proved that for any Banach space each everywhere convergent Haar series with coefficients from this space is the Fourier–Haar series in the sense of Henstock type integral with respect to a dyadic differential basis. At the same time, the almost everywhere convergence of a Fourier–Henstock–Haar series of a Banach-space-valued function essentially depends on properties of the space.     

Selection principles for maps of several variables

We show that a pointwise precompact sequence of maps from the n-dimensional rectangle into a metric semigroup, whose total variations in the sense of Vitali, Hardy and Krause are uniformly bounded, contains a pointwise convergent subsequence. We present a variant of this result for maps with values in a reflexive separable Banach space with respect to the weak pointwise convergence of maps.