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:

I am working on using the Morse covering theorem to define the Henstock/Kurzweil integral in finite dimensional spaces.

References for Henstock/Kurzweil integration.

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 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.

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.

Preprints