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Research- Erik Talvila

 

Henstock-Kurzweil integration

The Henstock-Kurzweil integral is an integral with a simple definition in terms Riemann sums but it includes the Riemann, Lebesgue, improper Riemann and Cauchy-Lebesgue 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 n-dimensional Euclidean space and in metric and topological spaces.

References for Henstock-Kurzweil 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), 491-496). The class of primitives is also known for Henstock-Kurzweil integrals. However, it is more complicated than the absolutely continuous functions. A problem with the Henstock-Kurzweil 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 Henstock-Kurzweil 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 Henstock-Kurzweil 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.

 

Fourier transforms

Because of their oscillatory kernel, it is natural to treat Fourier transforms as Henstock-Kurzweil integrals. This provides an extension of the Lebesgue theory. Some results are similar to the absolutely convergent case, such as an inversion theorem and convolution. But there are new phenomena such as arbitrarily large growth of the transform and the failure of the transform to exist on countable sets.

 

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 Henstock-Kurzweil sense on the circle.

 

Phragmen-Lindelof 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 Phragmen-Lindelof principles that allow the solution to blow up at the boundary but still yield uniqueness.

 

Publications

 

Talks

  • 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 mini-conference 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, ``Henstock-Kurzweil 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 Henstock-Kurzweil 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 Henstock-Kurzweil integral to the half plane Dirichlet problem''
  • Spring Miniconference in Real Analysis, California State University at San Bernardino, March 22, 2002, ``Henstock-Kurzweil Fourier transforms''
  • American Mathematical Society Special Session in Real Analysis, University of Tennessee, Chattanooga, TN, October 5, 2001, ``Pointwise Fourier inversion without the Riemann-Lebesgue 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 Urbana-Champaign. Colloquium. May 3, 2001 ``A survey of nonabsolute integration''

American Mathematical Society Special Session on Nonabsolute integration, Toronto, September 23-24, 2000.  

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