Non-absolutely Convergent Integrals Research Articles

Non-absolutely Convergent Generalized Laplacian

For possibly discontinuous functions including, for instance, Sobolev functions, we present new Blaschke-Privaloff-type criteria for superharmonicity and harmonicity. This opens the way for introduction of a substantial generalization of the Laplace operator. These potential-theoretic considerations lead to a new kind of non-absolutely convergent integral where the integrand may be a highly oscillating pointwise function or even a distribution-valued function. In turn, this integral gives a precise meaning to some generalized Poisson equations with a wild right hand side.

DIFFERENTIAL INCLUSIONS, NON-ABSOLUTELY CONVERGENT INTEGRALS AND THE FIRST THEOREM OF COMPLEX ANALYSIS

In the theory of complex valued functions of a complex variable arguably the first striking theorem is that pointwise differentiability implies C∞ regularity. As mentioned in Ahlfors [Ah 78] there have been a number of studies [Po 61], [Pl 59] proving this theorem without use of complex integration but at the cost of considerably more complexity. In this note we will use the theory of non-absolutely convergent integrals to firstly give a very short proof of this result without complex integration and secondly (in combination with some elements of the theory of elliptic regularity) provide a far reaching generalization. One of the first and most striking theorems about the analysis of complex valued functions of a complex variable is that merely from considering the class of pointwise complex differentiable functions on an open set we instantly find ourself in the category of C∞ functions. Theorem 1. Given open set Ω ⊂ C. Suppose f : Ω → C is a complex differentiable at every point. Then f is C∞ on Ω. Typically Theorem 1 is proved via the method of complex integration. The first step is to prove that the integral of a differentiable function over the boundary of a rectangle inside a ball is zero, this was first proved by Goursat [Go 01]. The existence of an anti-derivative is then concluded, Cauchy’s integral formula follows and it is shown that you can differentiate through the integral of Cauchy’s integral formula infinitely many times and hence the function is C∞. On the first paragraph of page 101 of Ahlfors’s standard text [Ah 78], he writes that many important properties of analytic functions are difficult to prove without use of complex integration. Ahlfors states that only recently1 it has been possible to prove continuity of the gradient (or the existence of higher gradients) without the use of complex integration. He refers to articles of Plunkett [Pl 59], and Porcelli and Connell [Po 61] both of which rely on a topological theorem of Whyburn [Wh 58]. Ahlfors notes that both these proofs are much more complicated than the original proof. It turns out that generalizations of Goursat’s theorem have a long history and one line of generalization provides an alternative proof of the theorem that in essence does not require complex integration. This line of research was started by Montel [Mo 07] and further developed by Looman [Lo 23] and Menchoff [Me 36] (their theorem receives a very clear exposition in Saks [Sa 37]) and later by Tolstov [To 42]. Although not explicitly stated in these works the method of proof was essentially to construct a Denjoy type integral to integrate the divergence of a differentiable vector field [Pr 12]. The explicit application of the theory of non-absolutely convergent integrals appears to have first been made in [Ju-No 90] by Jurkat and Nonnenmacher, they used a kind of Perron integral in the plane to prove a generalization of Goursat’s theorem due to Besicovitch [Be 31]. The purpose of this note is firstly to use the theory of non-absolutely convergent integrals to provide the shortest proof of Theorem 1, the proof we provide is also independent of the theory of complex line integrals. Secondly by rephrasing Theorem 1 in terms of differential inclusions 2010 Mathematics Subject Classification. 30A99,26A39,35J47.

Non-absolutely convergent integrals and singular integrals

We define the packing integral (a kind of non-absolutely convergent integral) with respect to distributions of arbitrary order. Then we show that singular integrals can be interpreted as packing integrals with respect to generating distributions. This allows us to consider singular integrals beyond \(L^1\).

Non-absolutely convergent integrals with respect to distributions

We define an integral of a function with respect to a distribution. In case that the underlying distribution is just the Lebesgue measure, the definition leads to a new non-absolutely convergent integral which is wider than the Denjoy–Perron integral. We present a version of the Gauss–Green theorem where the new integral is used for both interior and boundary terms. As a by-product, we characterize the predual Sobolev space \(W^{-1,1}\).

A CONSTRUCTIVE DEFINITION OF THE n-DIMENSIONAL ν(S)-INTEGRAL IN TERMS OF RIEMANN SUMS

In a former paper ([Ju-No 1]) we introduced an axiomatic approach to the theory of non-absolutely convergent integrals in \(\mathbb{R}^n\). A specialization of this abstract concept leads to the well-behaved \(\nu (S)\)-integral over quite general sets \(A\) which yields the divergence theorem in its presently most general form. (See [Ju-No 3].) While the definition of the \(\nu (S)\)-integral is of descriptive type (i.e. in terms of an additive almost everywhere differentiable set function) we prove in this paper that it can equivalently be defined using Riemann sums. As an application we show that any function being variationally integrable over \(A\) in the sense of [Pf 3] is also \(\nu (S)\)-integrable over \(A\) and both integrals coincide.

An axiomatic theory of non-absolutely convergent integrals in Rn

An axiomatic theory of non-absolutely convergent integrals in Rn

A theory of non-absolutely convergent integrals in Rn with singularities on a regular boundary

Specializing a recently developed axiomatic theory of non-absolutely convergent integrals in $ℝ^n$, we are led to an integration process over quite general sets $A ⊆ q ℝ^n$ with a regular boundary. The integral enjoys all the usual properties and yields t

Remarks on Nonabsolutely Convergent Integrals in the Real Line

Remarks on Nonabsolutely Convergent Integrals in the Real Line

A non-absolutely convergent integral which admits $C^1$-transformations

A non-absolutely convergent integral which admits $C^1$-transformations

On Mawhin's approach to multiple nonabsolutely convergent integral

On Mawhin's approach to multiple nonabsolutely convergent integral

The Representation of (C, k) Summable Series in Fourier Form

Several non-absolutely convergent integrals have been defined which generalize the Perron integral. The most significant of these integrals from the point of view of application to trigonometric series are the Pn- and pn-integrals of R. D. James [10] and [11]. The theorems relating the Pn -integral to trigonometric series state essentially that if the series1.1

On extensions of the Riemann and Lebesgue Integrals by Nets

In this note our principal interest is in using nets to give spaces of non-absolutely convergent integrals as extensions of the spaces of absolutely convergent Riemann and Lebesgue integrals. For this purpose we develop a general theory of extensions, by nets, of functions defined on the open intervals with closures in the complement of a fixed closed set, the nets being directed by inclusion for finite disjoint collections of such intervals. Two cases are considered leading to open extension (OE-) and conditional open extension (COE-) nets, the latter being subnets of the former. Necessary and sufficient conditions for the convergence of the OE- and COE-nets are given, those for the COE-nets being similar to conditions that arise in the definition of the restricted Denjoy integral. Properties of inner continuity, weak additivity and the existence of a continuous integral are defined and studied. These relate to the more specialized nets that are suitable for the extension of integrals.

The determination of the bulk stress in a suspension of spherical particles to order c 2

An exact formula is obtained for the term of order c2 in the expression for the bulk stress in a suspension of force-free spherical particles in Newtonian ambient fluid, where c is the volume fraction of the spheres and c [Lt ] 1. The particles may be of different sizes, and composed of either solid or fluid of arbitrary viscosity. The method of derivation circumvents the familiar obstacle, of non-absolutely convergent integrals representing the effect of all pair interactions in which one specified particle takes part, by the judicious use of a certain quantity which is affected by the presence of distant particles in a similar way and whose mean value is known exactly. The bulk stress is in general of non-Newtonian form and depends on the statistical properties of the suspension which in turn are dependent on the type of bulk flow.The formula contains two functions which are parameters of the flow field due to two spherical particles immersed in fluid in which the velocity gradient is uniform at infinity. One of them, p(r, t), represents the probability density for the vector r separating the centres of the two particles. The variation of p(r, t) for a moving material point in r-space due to hydrodynamic action is found in terms of a function q(r), and this gives p(r, t) explicitly over the whole of the region of r-space occupied by trajectories of one particle centre relative to another which come from infinity. In a region of closed trajectories, steady-state hydrodynamic action alone does not determine the relation between the values of p (r, t) for different material points. The function q(r) is singular when the spheres touch, and the contribution of nearly-touching spheres to the bulk stress is evidently important. Approximate numerical values of all the relevant functions are presented for the case of rigid spherical particles of uniform size.In the case of steady pure straining motion of the suspension, all trajectories in r-space come from infinity, the suspension has isotropic structure and the stress behaviour can be represented (to order c2) in terms of an effective viscosity . It is estimated from the available numerical data that for a suspension of identical rigid spherical particles \[ {\mathop\mu\limits^{*}}/\mu = 1 + 2.5c + 7.6c^2, \] the error bounds on the coefficient of c2 being about ∓ 0.8. In the important case of steady simple shearing motion, there is a region of closed trajectories of one sphere centre relative to another, of infinite volume. The stress system is here not of Newtonian form, and numerical results are not obtainable until the probability, density p(r, t) can be made determinate in the region of closed trajectories by the introduction of some additional physical process, such as three-sphere encounters or Brownian motion, or by the assumption of some particular initial state.In the analogous problem for an incompressible solid suspension it may be appropriate to assume that for many methods of manufacture p(r, t) is uniform over the accessible part of r-space, in which event the solid suspension has ‘Newtonian’ elastic behaviour and the ratio of the effective shear modulus to that of the matrix is estimated to be 1 + 2·5c + 5·2c2 for a suspension of identical rigid spheres.

Constructive Definitions for Non-Absolutely Convergent Integrals

Constructive Definitions for Non-Absolutely Convergent Integrals

A non-absolutely convergent integral in $E_m$ and the theorem of Gauss

A non-absolutely convergent integral in $E_m$ and the theorem of Gauss

Functions of bounded variation and non-absolutely convergent integrals in two or more dimensions

Functions of bounded variation and non-absolutely convergent integrals in two or more dimensions

The equivalence of sequence integrals and non-absolutely convergent integrals

The equivalence of sequence integrals and non-absolutely convergent integrals

Non-absolutely convergent integrals with respect to functions of bounded variation

Introduction. Problems of great interest in real-variable theory are the analysis of the structure of a function whose derivative is finite, and the determination of a function when its derivative is given and is finite at each point. The study of these problems has led to a theory of non-absolutely convergent integrals.t The corresponding problems when the derivative in question is with respect to a function of bounded variation have received little attention, and it is with these that the present paper is mainly concerned. In this case we find that there is involved a theory of non-absolutely convergent integrals with respect to a function of bounded variation. Lebesguet has given these questions some consideration. His results depend, in part, on a transformation which changes integration with respect to a function into ordinary integration. By means of this transformation very general results are easily obtained. But for dealing with some particular situations it is not altogether suitable; for example, a discussion of integral equations which involve integration with respect to a function of bounded variation. For this reason we have, in the present work, adhered to methods which are direct. Incidental to our main purpose is a study of the derivative of a function F(x) with respect to a function of bounded variation a(x). Here, too, we have been preceded by others, chiefly Daniell.? In the Transactions paper Daniell concerns himself with the central derivative with respect to a,

A Convergence Theorem for Non-Absolutely Convergent Integrals

A Convergence Theorem for Non-Absolutely Convergent Integrals

Concerning Harnack’s theory of improper definite integrals

In this paper I consicler the improper simple definite integrals of HARNACK (1883, 1884). In the introduction I wish to characterize somuewhat clearly the theories of the improper simple and multiple integrals recently given by JORDAN (1894) and STOLZ (1898, 1899), and in this introductory paragraph I summarize the contents of the whole introduction. These theories for the simple integrals have intimate relations with the HARNACK theory. The definition adopted for the multiple integrals is inore exacting than that for the simple ilntegrals. The miiultiple integrals converge or exist (as limits) only absolutely. For the simple integrals we have then two theories, on the one hanld, of the integrals with the milder definition, and, on the other hand, of the integrals with the stronger definition and so with a larger body of properties. The first class of integrals includes the second class of integrals. The HARNACK theory relates to the first and general class of integrals; this theory has not received systellmatic development; however, for the theory of the absolutely conivergent HARNACK integrals this is nlot true, and these initegrals conistitute the second and special class of integrals. I discuss both classes of simple integrals simultaneously and by uniform process; this is made possible by suitable determinations of the definitions; the absolute convergence of the integrals of the seconid class appears only at the conclusion, and hence it is desirable to introduce terms of discriminiation conlnoting the two definitions, the milder ancl the stronger; the terms chosen, narrow, broad, connote the geometric form of the definitions, and likewise the fact that the class of narrow integrals lhas a less extensive body of properties than the (included) class of broad integrals. There has been a tendency to do away with the non-absolutely convergent HARNACK integrals; I hope to show that this tendency rests uponi misconceptions.-The tlheory of DE LA VALL-fE POUSSIN (initiated in 1892) is in form distinct from the HARNACK theory and