Prof. dr. Ragnar Sigurðsson: Growth estimates of entire functions of finite order and type along certain complex lines in C^n with applications to Fourier analysis
Source: Mathematics colloquium
University of Iceland
The main motivation for the study of entire functions of exponential type in Cn is the fact that they are Fourier-Laplace transforms of functions with compact support, distributions with compact support, and analytic functionals. The relation between the convex hull of the support of a function or a distribution and growth estimates of its Fourier-Laplace transform is expressed in the Paley-Wiener theorems, which exist in many variants.
In the lecture I will discuss the following problem: Assume that we have a function defined on a set of complex lines through the origin in Cn and that its restriction to each of these lines is of finite order and finite type viewed as a function of one complex variable. Under which conditions can we conclude that the function can be extended from the union of lines to an entire function of finite order and finite type in the whole space Cn?
I will give sufficient condtions for a solution of the problem and show how the results can be applied for relaxing conditions in Paley-Wiener theorems. The methods I use are completeley elementary and only involve estimates of polynomials and power series, homogeneous functions, norms and a little bit of convexity theory.
(The lecture is based on a joint work with Jöran Bergh, Chalmers University of Technology, Gothenburg, Sweden.)