This talk is about work done in conjunction with Werner Kratz concerning the known theorem that a real sequence (s_n) which is summable by the Borel method, and which satisfies the one-sided Tauberian condition that Ön(s_n-s_n-1) is bounded below, must be convergent. We established a quantatitive version of Vijayaraghavan's classical result and used it to supply a short new proof of this Tauberian theorem.

We consider functions mapping a closed, bounded interval into an ordered vector space. Using the notion of order convergence, we define lim sups, lim infs, derivates, and upper and lower Henstock integrals for this type of function.

It is shown that some familiar facts from analysis generalize to this setting. For example, the derivative operator is linear, and the integral is a positive linear operator on the integrand, and is additive as a function on intervals. However, other familiar ideas, such as the Intermediate Value Theorem and the Mean Value Theorem, do not generalize to this setting.

I will discuss a two parameter family of fractals.

By a derivability theorem we mean a theorem which establishes derivability in a given sense of a certain class of functions at some points. After the shocking discovery of Weierstrass that there are continuous functions which are not derivable at any point, the first derivability theorem was obtained indeed by Lebesgue, namely that every function of bounded variation is differentiable almost everywhere.

In this talk we will present several derivability theorems in terms of some new derivatives, namely lower, upper and semi-derivatives, and their normalized versions. These derivability theorems hold for every continuous function.

Let f be a real (or complex) function with domain D_f containing the positive integers. We introduce the functional sequence {f_sn(x)} as follows:

In this talk, we introduce and discuss the topic of limit summability of real functions. Then focus on convex and concave limit summable functions.

(joint work with Chongsuh Chun and Hunnam Kim)

Recently, a series of basic principles of the usual measure theory such as Brooks-Jeweet theorem, Nikodym convergence theorem and Vitali-Hahn-Saks theorem also established for topological group valued measures defined on quantum logics. Non-commutative measure theory consists in replacing Boolean algebras by quantum logics such as orthoalgebras, effect algebras or D-posets. Note that effect algebras are a natural generalization of Boolean algebras and orthoalgebras, while D-posets are mathematical equivalent objects to effect algebras However, the usual theory of locally convex space valued measure is quite plentiful so it is necessary to consider vector measure defined on quantum logics. Now we establish boundedness results for locally convex space valued measures on effect algebras.

A combined effort of three important and influential papers in the 40's and 50's (N. Dunford and B.J. Pettis, Linear operations on summable functions. Trans. Amer. Math. Soc. 47(1940), 323-392; A. Grothendieck, Sur les applications lineaires faiblement compactes d'espaces du type C(K). Canad. J. Math. 5(1953), 129-173; R.G. Bartle, N. Dunford, and J.T. Schwartz, Weak compactness and vector measures.) demonstrated that every weakly compact bounded linear transformation (= operator) on the classical functions spaces L₁[0,1] and C[0,1] map weakly Cauchy sequences into norm convergent sequences. Grothendieck formalized this property as follows: A Banach space X has the Dunford-Pettis property (DPP) provided that every weakly compact operator with domain X and range an arbitrary Banach space Y is completely continuous; i.e., a weakly compact operator maps weakly compact sets in X into norm compact sets in Y. Localizing this notion, a bounded subset A of X is said

Theorem. If X is a Banach space which contains a non-relatively compact strong Dunford-Petts set, then c₀ \hookrightarrow K(X, X) and l_¥ \hookrightarrow L(X, X).

Theorem. The Banach space l₁ does not embed in X if and only if every Dunford-Pettis subset of X^* is relatively compact. The Banach space l₁ embeds complementably in X if and only if there is a strong Dunford-Pettis subset of X^* which is not relatively compact.

Let r(s) be a fixed infinitely differentiable function defined on R⁺=[0,¥) having the properties:

(i) r(s) ³ 0,
(ii) r(s) = 0 for s ³ 1,
(iii) ò_R^md_n (x) dx = 1

In this joint work with Erik Talvila, it is shown that the approximating functions used to define the Bochner integral can be formed using geometrically nice sets, such as balls, from a differentiation basis. Moreover, every apropriate sum of this form will be within a preassigned e of the integral. All of this follows from the ubiquity of Lebesgue points, which is a consequence of Lusin's theorem, for which a simple proof is included in the discussion.

A Cantor set is a compact, totally disconnected and perfect subset of the real numbers. Such a set is completely determined by its "gaps". We suggest a method of associating a Cantor set with a summable sequence of positive numbers and investigate the relationship between the asymptotics of this sequence and the Hausdorff measure and dimension of the resulting set.

For example, if a ~ b (the sequences are asymptotic), then the resulting Cantor sets have the same dimension.

In the particular case of the sequence l_p = 1/n^p, it is known that the dimension of the associated Cantor set is 1/p. Using this as a starting point, we compare the asymptotics of a given sequence a to l_p using various measures of "asymptotics".

Finally, we show that for any non-increasing sequence, the resulting Cantor set has positive and finite Hausdorff h-measure, where h is a dimension function that is naturally associated with the sequence.

This work is joint work with Carlos Cabrelli, Ursula Molter and Ron Shonkwiler.

Riemann integration in one dimension or n dimensions involves partitioning the domain of integration. The generalized Riemann integral of Henstock and Kurzweil requires that such partitions satisfy certain conditions, and the existence of the partitions is guaranteed by Cousin's Lemma. Problems in stochastic analysis involve integration in infinite dimensional domains, and suitable partitions of these domains are needed for a generalized Riemann approach to these problems.

We develop the L^r-Henstock-Kurzweil (HK_r) Integral, which extends the integral of Henstock and Kurzweil to integrate all L^r-derivatives, and which employs a Riemann-type construction. We show that the HK_r integral extends the P_r integral of L. Gordon. We give a condition analogous to that of absolute continuity to characterize functions which are HK_r integrals. Finally we give convergence theorems which hold for sequences of HK_r integrable functions. This is joint work with Yoram Sagher.

Let (G, +) be an additive subgroup of the reals. A subset S of R is said to be k difference free if for every nonzero g in G the equation g=x-y has less than k solutions in S. In this talk we discuss some results concerning difference free sets and sum free sets. Among other things, we show that for any proper additive subgroup H of the reals, P+H is not residual in R for any finite difference free set P, but A+H=R for some set A that has no arithmetic sequence of length three. An application of one of our results concerning sum free sets is the following: For any function f from the reals to a finite set, the set of all x such that {h > 0: f(x-h)=f(x+h)} is infinite is of the size of the continuum.

Several different classes of functions arise naturally in the study of the convergence of Fourier series. We concern ourselves here with two such classes: LBV and FBV, which are classes of functions of generalized bounded variation. These classes have their origins in the work of L.C. Young, and have been developed extensively by D. Waterman. We show here that the necessary and sufficient condition for g °f to be in the class FBV, LBV for every f of that class whose range is in the domain of g is that g be in Lip1.

In this paper we give a general Riemann-type integration theory, which includes strictly the Bochner integral. Our theory is a natural countable extension of the abstract Riemann integral theory defined by us in a recent paper.

The generalized Riemann integral that we define using the generalized Riemann sums concerns the functions f: (T,S,m)® B, where T is an abstract set, S is a sigma-ring of parts of T, m: S®[0,®] is a countable additive measure and B is a Banach space. In the particular case when T=B=R and m is the Lebesgue measure, our integral coincides with the classic Lebesgue integral. In the general case, the set of Bochner integrable functions is strictly included in the set of generalized Riemann m-integrable functions.

The theory we give is similar to the classic Riemann integral theory. In the context of the abstract Lebesgue integral theory new results are emphasized. Such one is the characterization of the Lebesgue m-integrability by the generalized Darboux m-integrability, concept we define by natural countable extension of the concept of classic Darboux integrability.

By this paper, in the particular case of real functions, we solve the open problem of defining the Lebesgue integral as Riemann-type integral.

Lyapunov's Theorem asserts that the range of an atomless vector measure is convex. Early proofs of this results were obscure and fiddly; more recent short proofs use relatively heavy machinery from convexity theory. I will give an entirely elementary, very short proof of the theorem, which obtains the result as a consequence of the Intermediate Value Theorem.

We will discuss mixing rank one transformations, and in particular a theorem that proves a rank one transformation satisfying a condition called restricted growth is a mixing transformation if and only if the spacer sequence for the transformation is uniformly ergodic. Uniform ergodicity is a generalization of the notion of ergodicity for sequences, in the sense that the mean ergodic theorem holds for a family of what we call dynamical sequences. The application of our theorem shows that the class of polynomial rank one transformations, rank one transformations where the spacers are chosen to be the values of a polynomial with some mild conditions on the polynomials, that have restricted growth are mixing transformations, implying in particular Adams' result on staircase transformations. Another application yields a new proof that Ornstein's class of rank one transformations constructed using "random spacers" are almost surely mixing transformations. This is joint work with Darren Creutz.

In this paper we establish the various relationships that exist among the integral transform, the convolution product, and the first variation for a class of functionals defined on K[0,T], the space of complex-valued continuous functions y(t) on [0,T] which vanish at t=0.

If f:R®R is Henstock-Kurzweil integrable then the Alexiewicz norm is ||f||=sup_{I Ì R}|ò_I f| where the supremum is taken over all intervals I Ì R. The resulting normed linear space is not complete. Its completion is the set of Schwartz distributions that are the distributional derivative of a bounded continuous function that vanishes at -¥. This describes the distributional Denjoy integral on R. On Rⁿ there are two natural extensions of this definition. The first leads to an integral that has a very strong divergence theorem. The second gives an integral that inverts the nth order mixed partial derivative. These integrals have simple definitions and yet extend many of the nonabsolute integrals to Rⁿ in an effective manner.

Given a space, X, a function r: X×X®R⁺ is called a quasi-metric if (a) r(x,y) = r(y,x) = 0 if and only if x=y, and (b) for every x,y,z Î X, r( x,y) £ r( x,z) +r(z,y). Furthermore, a quasi-metric space ( X,r) is weightable if there exists a function w:X® R ⁺ such that

We provide a porosity notion approach to the differentiability and continuity of real valued functions on separable Banach spaces, when the function is monotone with respect to an ordering induced by a convex cone K with non-empty interior. We also show that the set of nowhere K-monotone functions has a s-porous complement in the space of the continuous functions.

(Joint work with T.L. Gill, S. Basu and V. Steadman.)

It is shown that a result of L. Gross and J. Kuelbs, used by several authors to study Gaussian measures on Banach spaces, makes it possible to construct an adjoint for operators on separable Banach spaces. This result is then used to extend well known results of J. von Neumann and P.D. Lax. We also partially solve an open problem on the existence of a Markushevich basis with unit norm and prove that all closed densely defined linear operators on a separable Banach space can be approximated by bounded operators. The latter result extends a theorem of W.E. Kaufman for Hilbert spaces and allows us to define a new metric for closed densely defined linear operators on separable Banach spaces. As an application, we obtain a generalization of the Yosida approximator for operator semigroups.