Ph.D. (Waterloo), M.Sc. (Western Ontario), B.Sc. (Toronto)
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.
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.
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.
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.
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.
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.
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.
UFV student authors underlined. See the arxiv for preprints.
- Erik Talvila, The continuous primitive integral in the plane, Real Analysis Exchange, 20(2020), 283--326.
- Erik Talvila, Fourier transform inversion using an elementary differential equation and a contour integral, American Mathematical Monthly, 126(2019), 717--727.
- Erik Talvila, Higher order corrected trapezoidal rules in Lebesgue and Alexiewicz spaces, Journal of Classical Analysis, 8(2016), 77--90.
- Erik Talvila, The one-dimensional heat equation in the Alexiewicz norm, Advances in Pure and Applied Mathematics, 6(2015), 13--37.
- Erik Talvila, The $L^p$ primitive integral, Mathematica Slovaca, 64(2014), 1497--1524.
- Seppo Heikkila and Erik Talvila, Distributions, their primitives and integrals with applications to distributional differential equations, Dynamic Systems and Applications, 22(2013), 207--249.
- Erik Talvila, Trigonometry of The Gold-Bug, Mathematical Gazette, 97, Number 538, March 2013, 124--127.
- Erik Talvila and Matthew Wiersma, Optimal error estimates for corrected trapezoidal rules, Journal of Mathematical Inequalities, 6(2012), 431--445.
- Erik Talvila and Matthew Wiersma, Simple derivation of basic quadrature formulas, Atlantic Electronic Journal of Mathematics, 5(2012), 47--59.
- Erik Talvila, Integrals and Banach spaces for finite order distributions, Czechoslovak Mathematical Journal 62 (2012), 77-104.
- Erik Talvila, Fourier series with the continuous primitive integral, Journal of Fourier Analysis and Applications 18 (2012), 27-44.
- Erik Talvila, The regulated primitive integral, Illinois Journal of mathematics 53 (2009), 1187-1219.
- 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), 51-82.
- Erik Talvila, Continuity in the Alexiewicz norm, Mathematica Bohemica 131 (2006), 189-196.
- Erik Talvila, Estimates for Henstock-Kurzweil Poisson integrals, Canadian Mathematical Bulletin 48 (2005), 133-146.
- Erik Talvila, Estimates of the remainder in Taylor's theorem using the Henstock-Kurzweil integral, Czechoslovak Mathematical Journal 55(130) (2005), 933-940.
- Peter A. Loeb and Erik Talvila, Lusin's Theorem and Bochner integration, Scientiae Mathematicae Japonicae 60 (2004), 113-120.
- Parasar Mohanty and Erik Talvila, A product convergence theorem for the Henstock-Kurzweil integral, Real Analysis Exchange 29 (2003/2004), 199-204.
- Erik Talvila, Henstock-Kurzweil Fourier transforms, Illinois Journal of Mathematics 46 (2002), 1207-1226.
- David Siegel and Erik Talvila, Sharp growth estimates for modified Poisson integrals in a half space, Potential analysis 15 (2001) 333-360.
- Erik Talvila, Rapidly growing Fourier integrals, American Mathematical Monthly, 108 (August-September 2001) 636-641.
- Erik Talvila, Necessary and sufficient conditions for differentiating under the integral sign, American Mathematical Monthly, 108 (June-July 2001) 544-548.
- Erik Talvila, Some divergent trigonometric integrals, American Mathematical Monthly 108 (May 2001) 432-436.
- Peter A. Loeb and Erik Talvila, Covering theorems and Lebesgue integration, Scientiae Mathematicae Japonicae 53 (2001) 91-103.
- Erik Talvila, Limits and Henstock integrals of products, Real Analysis Exchange 25 (1999/00) 907-918.
- David Siegel and Erik Talvila, Pointwise growth estimates of the Riesz potential, Dynamics of Continuous, Discrete and Impulsive Systems 5 (1999) 185-194.
- David Siegel and Erik Talvila, Uniqueness for the n-dimensional half space Dirichlet problem, Pacific Journal of Mathematics 175 (1996) 571-587.
- Erik Talvila, Growth estimates and Phragmen-Lindelof 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.