Research Erik Talvila
HenstockKurzweil integration
The HenstockKurzweil integral is an integral with a simple definition in terms
Riemann sums but it includes the Riemann, Lebesgue, improper Riemann
and CauchyLebesgue integrals. It has the following advantages over
Riemann and Lebesgue integrals:
 An elementary definition that requires no knowledge of measure
theory produces an integral more general than the Lebesgue integral.
 It is nonabsolute. A function can be integrable without its absolute
value being integrable.
 Every derivative is integrable. This property is not held by
Riemann or Lebesgue integrals! We thus get the most complete version of
the Fundamental Theorem of Calculus and the divergence theorem.
 The integral can be defined with respect to finitely additive
measures in ndimensional Euclidean space and in metric and topological spaces.
References for HenstockKurzweil integration.
Distributional integrals
A convenient way to define an integral is through properties of its
primitive. The primitive is a function whose derivative is in some
sense equal to the integrand. For example, in Lebesgue integration
the primitives are the absolutely continuous functions. A function
f
on the real line is integrable in the Lebesgue sense if and only if there is
an absolutely continuous function F such that F'=f
almost everywhere. For integration on the entire real line, the
primitive must also be of bounded variation. Primitives for Riemann
integrals have recently been categorised by Brian Thomson
(Characterization of an indefinite Riemann integral,
Real Analysis Exchange 35 (2009/2010), 491496).
The class
of primitives is also known for HenstockKurzweil integrals. However,
it is more complicated than the absolutely continuous functions. A
problem with the HenstockKurzweil integral is that the space of
integrable functions is not a Banach space. By
taking the primitives as continuous functions and using the distributional
derivative we obtain the continuous primitive integral. This
includes the Lebesgue and HenstockKurzweil integrals. Under the
Alexiewicz norm, the space of distributions integrable in this sense
is a Banach space. It is the completion of the Lebesgue and
HenstockKurzweil integrable functions.
See the paper below. Primitives can also be
taken as regulated functions, i.e., those that have a left and a right
limit at each point. Or they can be taken as Lp functions.
Numerical integration
A simple integration by parts technique produces quadrature formulas
for the integral of a function of one variable with error in terms of
the Lp norm of some derivative of the function. Sharp error
estimates are obtained by allowing derivative terms to appear, evaluated
at the ends of the integration interval. This method also leads to
elementary derivations of the trapezoidal, midpoint and Simpsons rules.
Fourier series
Using the Alexiewicz norm, Fourier series with the continuous primitive
integral have many of the same properties as Fourier series have when the
Lebesgue integral is used with the L^1 norm. See the paper
Fourier series with the continuous primitive integral below.
Poisson integrals
The Poisson integral solves the classical Dirichlet problem for the
Laplace equation in a half space but existence
of the integral imposes certain growth restrictions on the boundary data.
It is possible to form a modified Poisson integral by subtracting terms
from the Taylor/Fourier expansion of the Poisson kernel. This then lets
us write the solution of the Dirichlet problem for arbitrary locally
integrable data. I have been working on obtaining
the best pointwise and norm estimates of these modified Poisson integrals.
Poisson integrals have been considered in the HenstockKurzweil sense on
the circle.
PhragmenLindelof principles
An elliptic partial differential equation in a bounded domain
will have a unique solution if boundary data is specified, provided
the coefficients, boundary and boundary data are reasonably well
behaved. For an unbounded domain we need some sort of growth condition
at infinity to be imposed in order to have a unique solution. I am
interested in PhragmenLindelof principles that allow the solution to
blow up at the boundary but still yield uniqueness.
Expository Writing
Publications
 Erik Talvila,
The onedimensional heat equation in the Alexiewicz norm,
Advances in Pure and Applied Mathematics,
(accepted January 16, 2015).
 Erik Talvila,
The L^p primitive integral,
Mathematica Slovaca, 64(2014), 14971524.
 Seppo Heikkila and Erik Talvila,
Distributions, their primitives and integrals with applications
to distributional differential equations,
Dynamic systems and applications,
22(2013), 207249.
 Erik Talvila,
Trigonometry of the GoldBug,
Mathematical Gazette, 97, Number 538,
March 2013, 124127.
 Erik Talvila and Matthew Wiersma,
Optimal error estimates for corrected trapezoidal rules,
Journal of Mathematical
Inequalities, 6(2012), 431445.
 Erik Talvila and Matthew Wiersma,
Simple derivation of basic quadrature formulas,
Atlantic electronic journal of mathemtatics
5(2012), 4759.
 Erik Talvila,
Integrals and Banach spaces for finite order distributions,
Czechoslovak Mathematical Journal
62 (2012), 77104.
 Erik Talvila,
Fourier series with the continuous primitive integral,
Journal of Fourier Analysis and Applications
18 (2012), 2744.
 Erik Talvila,
The regulated primitive integral,
Illinois Journal of mathematics 53 (2009), 11871219.
 Erik Talvila,
Convolutions with the continuous primitive integral,
Abstract and Applied Analysis (2009), Art. ID 307404, 18 pp.
Corrected version
 Erik Talvila,
The distributional Denjoy integral,
Real Analysis Exchange 33 (2008), 5182.
 Erik Talvila,
Continuity in the Alexiewicz norm,
Mathematica Bohemica 131 (2006), 189196.
 Erik Talvila,
Estimates for HenstockKurzweil Poisson integrals,
Canadian Mathematical Bulletin
48 (2005), 133146.
 Erik Talvila,
Estimates of the remainder in Taylor's
theorem using the HenstockKurzweil integral,
Czechoslovak Mathematical Journal
55(130) (2005), 933940.
 Peter A. Loeb and Erik Talvila,
Lusin's Theorem and Bochner
integration,
Scientiae Mathematicae Japonicae
60 (2004), 113120.
 Parasar Mohanty and Erik Talvila,
A product convergence theorem
for the HenstockKurzweil integral,
Real Analysis Exchange
29 (2003/2004), 199204.
 Erik Talvila,
HenstockKurzweil Fourier transforms,
Illinois
Journal of Mathematics 46 (2002), 12071226.
 David Siegel and Erik Talvila,
Sharp growth estimates for modified
Poisson integrals in a half space, Potential analysis
15 (2001) 333360.
 Erik Talvila,
Rapidly growing Fourier integrals,
American Mathematical Monthly, 108 (AugustSeptember 2001) 636641.
 Erik Talvila,
Necessary and sufficient conditions for differentiating
under the integral sign,
American Mathematical Monthly, 108 (JuneJuly 2001) 544548.
 Erik Talvila,
Some divergent trigonometric integrals,
American Mathematical Monthly 108 (May 2001) 432436.
 Peter A. Loeb and Erik Talvila,
Covering theorems and Lebesgue integration,
Scientiae Mathematicae Japonicae 53 (2001) 91103.
 Erik Talvila,
Limits and Henstock integrals of products,
Real Analysis Exchange 25 (1999/00) 907918.
 David Siegel and Erik Talvila,
Pointwise growth estimates of the
Riesz potential, Dynamics of Continuous, Discrete and
Impulsive Systems 5 (1999) 185194.
 David Siegel and Erik Talvila,
Uniqueness for the ndimensional
half space Dirichlet problem,
Pacific Journal of Mathematics 175
(1996) 571587.
 Erik Talvila,
Growth estimates and PhragmenLindelof principles for
half space problems,
Ph.D. thesis, University
of Waterloo, Waterloo, 1997.
 Erik Talvila,
A finite Bessel transform,
M.Sc. thesis, University
of Western Ontario, London, 1991.
Talks

Math Club, University of the Fraser Valley, March 11, 2016,
``HenstockKurzweil Integration"

MAA session, Topics and Techniques for Teaching Real Analysis,
joint MAA/AMS meeting,
Seattle, January 6, 2016,
"Continuous functions on the extended real plane"

MAA session, Topics and Techniques for Teaching Real Analysis,
joint MAA/AMS meeting,
Baltimore, January 15, 2014,
"Applications of the Alexiewicz norm"

Analysis, Dynamics and Applications Seminar,
University of Arizona, Tucson,
November 19, 2013
"Distributional solutions of the heat equation"

Mathematics and Statistics Seminar,
University of the Fraser Valley,
November 12, 2012
"Distributions"

XXXVI Summer Symposium in Real Analysis,
Berks College, University of Pennsylvannia,
June 27, 2012
"Fourier series with the continuous primitive integral"

MAA session, Topics and Techniques for Teaching Real Analysis,
joint MAA/AMS meeting,
Boston, January 6, 2012,
"A simple derivation of the trapezoidal rule for numerical integration"

24th Auburn miniconference in harmonic analysis, Auburn University,
Auburn, Alabama, November 19, 2010
"Fourier series with the continuous primitive integral"

Colloquium on differential equations and integration theory,
Krtiny, Czech Republic, October 16, 2010,
"Distributional integrals"

XXXIV Summer Symposium in Real Analysis, College of Wooster, Wooster, Ohio,
July 14, 2010
"Convolutions with the continuous primitive integral"

PNW MAA Annual general meeting, Central Washington University, Ellensburg,
April 4, 2009,
"The continuous primitive integral"

XXXII Summer Symposium in Real Analysis,
Chicago State University, June 8, 2008,
"Banach lattice for distributional integrals"

MAA session, Topics and Techniques in Real Analysis, joint MAA/AMS meeting,
San Diego, January 7, 2008, "Distributional integrals"

XXX Summer Symposium in Real Analysis,
University of North Carolina, Asheville, June 2006,
"The regulated integral on the real line"

XXIX Summer Symposium in Real Analysis,
Whitman College, Walla Walla, Washington, June 22, 2005,
"Distributional integrals
on the real line''

11th Meeting on Real Analysis and Measure
Theory, Hotel Terme, Ischia, Italy, July 16, 2004,
"The Morse covering theorem and integration"

XXVIII Summer Symposium in Real Analysis,
Slippery Rock University, Slippery Rock, Pennsylvania,
June 2004,
"Covering Theorems and Integration"

American Mathematical Society,
University of Southern California, Los Angeles, April 3, 2004,
"Distributional integrals: descriptive and Riemann sum definitions"

Canadian Mathematical Society,
University of Alberta, University of Alberta, Edmonton, Alberta,
June 15, 2003,
"The distributional Denjoy integral''

University of Missouri at Kansas City, March 11, 2003, "HenstockKurzweil
Fourier transforms''

University of Waterloo, August 20, 2002, "Nonabsolutely convergent
Fourier transforms''

Washington and Lee University, Lexington, MA, XXVI Summer Symposium on
Real Analysis, June 26, 2002, "The Dirichlet problem with HenstockKurzweil boundary data''

University College of the Fraser Valley, June 6, 2002, "Asymptotics of
Fourier transforms''

American Mathematical Society Special Session in Potential Theory,
Universite de Montreal, May 4, 2002, "Application of the HenstockKurzweil integral to the half plane
Dirichlet problem''

Spring Miniconference in Real Analysis, California State University at
San Bernardino, March 22, 2002, "HenstockKurzweil Fourier transforms''

American Mathematical Society Special Session in Real Analysis, University of
Tennessee, Chattanooga, TN, October 5, 2001,
"Pointwise Fourier inversion without the RiemannLebesgue Lemma''

XXV SUMMER SYMPOSIUM IN REAL ANALYSIS, Weber State University, Ogden, Utah. May 26, 2001
"Half plane Dirichlet and Neumann problems''

University of Illinois at UrbanaChampaign. Colloquium. May 3, 2001
"A survey of nonabsolute integration''
American Mathematical Society Special Session on Nonabsolute integration,
Toronto, September 2324, 2000.
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