Mathematics and Statistics

# Faculty

Back to Faculty ### Erik Talvila

Associate Professor

Mathematics and Statistics

Abbotsford campus

## Education

Ph.D. (Waterloo), M.Sc. (Western Ontario), B.Sc. (Toronto)

## Research Interests

### 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.

### 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 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. 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 series

Because of their oscillatory kernel, it is natural to treat Fourier series as Henstock-Kurzweil or continuous primitive 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 convergence of the symmetric partial sums within the Alexiewicz norm.

### Numerical integration

A simple integration by parts technique leads to numerical integration algorithms in the form of the trapezoidal rule, midpoint rule and Simpson's rule.  Error estimates are given in terms of Lp or Alexiewicz norms.  Modified trapezoidal rules incorporating derivatives are shown to be optimal with respect to certain norms.   These methods also apply to multivariable integration.

### Heat equation

The heat equation is solved on the real line using the continuous primitive integral.  Initial conditions are taken on in the Alexiewicz norm.  Various sharp estimates are proved.

### 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 was once interested in Phragmen-Lindelof principles that allow the solution to blow up at the boundary but still yield uniqueness.

## Presentations

• ``The L^p primitive integral" October 23, 2019, International Online Seminar Mexico--Czech Republic--Canada--Brazil, Monterrey, Mexico.
• ``How large is a conditionally convergent series or integral" April 21, 2018, Mathematical Association of America, Seattle University.
• ``The heat equation with the continuous primitive integral" July 25, 2017, Mathematical Congress of the Americas, Montreal.
• ``What is an algorithm?" February 21, 2017, Microlecture, University of the Fraser Valley, Abbotsford.
• ``Henstock--Kurzweil integrals" March 11, 2016, Math Club, University of the Fraser Valley, Abbotsford.
• ``Continuous functions on the extended real plane" January 6, 2016, Mathematical Association of America, Seattle.
• ``Applications of the Alexiewicz norm" January 15, 2014, Mathematical Association of America, Baltimore.
• ``Distributional solutions of the heat equation" November 19, 2013, Applied Mathematics Seminar, University of Arizona, Tucson.
• ``Distributions" November 12, 2012, Math Club, University of the Fraser Valley, Abbotsford.
• ``Fourier series with the continuous primitive integral" June 27, 2012, XXXVI Summer Symposium in Real Analysis, Berks College, Pennsylvania State University, Reading, Pennsylvania.
• ``Research tools in mathematics and statistics" January 23, 2012, Math Club, University of the Fraser Valley, Abbotsford.
• 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.