


Real Analysis / Analyse Réel (Org: Erik Talvila)
 DAVID BORWEIN, Department of Mathematics, University of Western Ontario
Middlesex College, London, Ontario N6A 5B7
A onesided Tauberian theorem for the Borel summability method

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 onesided
Tauberian condition that Ön(s_{n}s_{n1}) 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.
 JOHN COFFEY, Purdue University Calumet, Hammond, Indiana 46323, USA
Differentiation and Henstock integration of functions taking
values in an ordered vector space

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.
 RICHARD DARST, Colorado State University
A fractal family

I will discuss a two parameter family of fractals.
 KRISHNA GARG, University of Alberta
Derivability theorems in terms of some new derivatives

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 semiderivatives, and their
normalized versions. These derivability theorems hold for every continuous
function.
 HADI HOOSHMAND, Department of Mathematics, Faculty of Natural Sciences,
Semnan University, Semnan, Iran
Convex and concave limit summable functions

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:
f_{sn}(x)=xf(n)+ 
n å
k=1


æ è

f(k)f(x+k) 
ö ø



and say that the function f limit summable at the point x_{0}
if the sequence {f_{sn}(x_{0})} is convergent,
(f_{sn}(x_{0})® f_{s}(x_{0})) as n®¥, and we call the function f_{s}(x) as the
limit summand function (of f).
In this talk, we introduce and discuss the topic of limit summability
of real functions. Then focus on convex and concave limit summable
functions.
 HONG TAEK HWANG, Kumoh, South Korea
Bounded vector measures on effect algebra

(joint work with Chongsuh Chun and Hunnam Kim)
Recently, a series of basic principles of the usual measure theory such
as BrooksJeweet theorem, Nikodym convergence theorem and
VitaliHahnSaks theorem also established for topological group valued
measures defined on quantum logics. Noncommutative measure theory
consists in replacing Boolean algebras by quantum logics such as
orthoalgebras, effect algebras or Dposets. Note that effect algebras
are a natural generalization of Boolean algebras and orthoalgebras,
while Dposets 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.
 PAUL LEWIS, University of North Texas, Denton, Texas 762031430, USA
Strong DunfordPettis sets and spaces of operators

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), 323392;
A. Grothendieck, Sur les applications lineaires faiblement
compactes d'espaces du type C(K). Canad. J. Math. 5(1953),
129173; 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 L_{1}[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 DunfordPettis 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 nonrelatively compact strong DunfordPetts set, then c_{0} \hookrightarrow K(X, X) and l_{¥} \hookrightarrow L(X, X).
Theorem. The Banach space l_{1} does not embed in X if and only if every DunfordPettis subset of X^{*} is relatively compact. The Banach space l_{1} embeds complementably in X if and only if there is a strong DunfordPettis subset of X^{*} which is not relatively compact.
 CHENKUAN LI, Brandon University, Brandon, Manitoba R7A 6A9
On defining the product r^{k} ·Ñ^{l} d

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_{Rm}d_{n} (x) dx = 1
where d_{n} (x)=c_{m}n^{m}r(n^{2}r^{2}) and c_{m} is the
constant satisfying (iii). In this talk, we overcome difficulties
arising from computing Ñ^{l} d_{n} and express this regular
sequence by two mutual recursions and use a Java swing program to
evaluate corresponding coefficients. Hence we are able to imply the
distributional product r^{k} ·Ñ^{l} d with the help
of Pizetti's formula and the normalization.
 PETER LOEB, University of Illinois, Urbana, Illinois 61801, USA
Lusin's Theorem and Bochner Integration

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.
 FRANKLIN MENDIVIL, Acadia University, Wolfville, Nova Scotia B4P 2R6
Hausdorff measure and dimension of general cantor sets

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 nonincreasing sequence, the resulting
Cantor set has positive and finite Hausdorff hmeasure, 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.
 PATRICK MULDOWNEY, University of Ulster, Northland Road, Derry BT48 7JL,
Northern Ireland
Cousin's Lemma in infinite dimensions

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.
 PAUL MUSIAL, Chicago State University, Chicago, Illinois 606281598, USA
The L^{r}HenstockKurzweil Integral

We develop the L^{r}HenstockKurzweil (HK_{r}) Integral, which
extends the integral of Henstock and Kurzweil to integrate all
L^{r}derivatives, and which employs a Riemanntype 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.
 KANDASAMY MUTHUVEL, Department of Mathematics, UWOshkosh, Oshkosh,
Wisconsin 549018631, USA
Sets with no repeated differences

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=xy 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(xh)=f(x+h)} is infinite is of the size of the continuum.
 PAMELA PIERCE, The College of Wooster
On the invariance of classes FBV, LBV under
composition

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.
 FLORIN POPOVICI, N. Titulescu College, Romania
A general Riemanntype integration theory including the
Bochner integral

In this paper we give a general Riemanntype 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 sigmaring 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
mintegrable 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
mintegrability by the generalized Darboux mintegrability, 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 Riemanntype
integral.
 DAVID ROSS, University of Hawaii
An elementary proof of Lyapunov's Theorem

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.
 CESAR SILVA, Williams College
Mixing on a class of rank one transformations

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.
 DAVID SKOUG, University of NebraskaLincoln, Lincoln, Nebraska 685880323,
USA
Integral transforms, convolution products, and first variations
of functionals on Wiener space

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
complexvalued continuous functions y(t) on [0,T] which vanish at
t=0.
 ERIK TALVILA, Alberta, Edmonton
The distributional Denjoy integral

If f:R®R is HenstockKurzweil 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 R^{n} 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
R^{n} in an effective manner.
 ROBERT VALLIN, Slippery Rock University of Pennsylvania, USA
On preserving quasimetrics and weightability

Given a space, X, a function r: X×X®R^{+} is called a quasimetric 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 quasimetric
space ( X,r) is weightable if there exists a function w:X® R ^{+} such that
r( x,y ) +w ( x ) = r( y,x ) +w (y ) . 

In this talk, we look at the set of f:R^{+}®R ^{+} such that f°r is a quasimetric on
X whenever r is. These f will be shown to be precisely the
metric preserving functions already in the literature. We then turn our
attention to the metric preserving functions which preserve the
property of weightability.
 SHAWN WANG, Okanagan University College
Cone monotone functions: differentiability and continuity

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 nonempty interior. We also show that the set of
nowhere Kmonotone functions has a sporous complement in the
space of the continuous functions.
 WOODFORD ZACHARY, Howard University, Washington, DC, USA
An adjoint for operators in Banach spaces

(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.

