Gæsteforedrag ved C.J. Chapman, University of Keele, UK og Benjamin Buus Støttrup, Aalborg Universitet


17.09.2021 kl. 13.00 - 15.30


13:00- 13:50 C.J. Chapman, University of Keele, UK

The finite product method in approximation theory, and some applications

Many well-known functions in mathematics can be written as infinite products of simple  factors. These include all the basic functions of trigonometry, of which Euler's infinite product  for the sine is the best known. Unfortunately, truncations of these expressions to finite products  are not normally of use, because of Runge's phenomenon, which is the presence of enormous  unwanted oscillations near the boundaries of the domain of interest. In this talk, it will be shown that in a class of applied problems in wave propagation, these highamplitude oscillations cancel out exactly, to leave an extremely useful family of finite-product  approximations, whose high accuracy and range of validity are extraordinary. The talk includes  a full account of Runge's phenomenon (for researchers new to the topic), a simple proof of the  exact cancellation, using only Stirling's approximation to the Gamma Function (with the `onetwelfth correction'), and some examples of wave propagation in which the resulting finiteproduct approximations have been put to good use by the speaker and Professor S. V. Sorokin 

14:00 - 14:30 coffee and cake 

14:30- 15:20 B.B. Støttrup, Aalborg Universitet, DK


The effectiveness of a one-dimensional acoustic black hole has often been characterized by its reflection coefficient R and its so-called normalized wave number variation, both of which depend on the height profile of the beam. The underlying assumptions of existence of acoustic black holes require the normalized wave number variation to be small. In this work, we consider an acoustic black hole with fixed parameters of geometry (i.e. length, maximal and minimal height) and pose the variational problem of finding a height profile which minimizes R under the additional constraint of keeping L2n norm of the normalized wave number variation small, since for large n this norm approximates a point-wise estimate. Using the method of Lagrange multipliers, we solve this variational problem and prove that the optimal profile is one of three different types. These profile types can be seen as direct generalizations of the well researched profile h(x) = x2 . In the limiting case n → ∞, we obtain closed form expressions for the optimal profiles as well as the corresponding reflection coefficient and the normalized wave number variation. Furthermore, we compare numerically the performance of the optimal profiles and profiles previously considered in the literature. In this comparison, we demonstrate that for comparable reflection coefficients the normalized wave number variation of the optimal profile is far superior to the other profiles.

This talk is based on joint work with Professor Sergey Sorokin from the Department of Materials and Production and Professor Horia Cornean from the Department of Mathematical Sciences


Department of Mathematical Sciences


Skjernvej 4A, AUD B