Division of Applied Mathematics, Mälardalen University, Västerås, Sweden

Biography: Dr Richard Bonner is Associate Professor at the Division of Applied Mathematics, Mälardalen University, Västerås, Sweden. After completing his PhD in Mathematics (systems of differential and convolution equations in several variables) at the University of Stockholm in 1984, he held a two-year postdoc position at the Mittag-Leffler Institute, Stockholm, and spent the following two years as researcher at the Centre for Mathematical Analysis, Australian National University, Canberra, and the University of Queensland, Brisbane. His interests then shifted towards border areas of Computing and Economics, spending eight years as senior lecturer in Economic Informatics at Griffith University, Brisbane, focusing on the (mathematical) methods of Artificial Intelligence. After returning to Sweden seventeen years ago to take up his current position, he became involved in the theories of quantum computation and headed international research projects that effectively initiated the area of quantum learning.

Speech Title: Can one tell a signal by its norms?

Abstract: A nonnegative function on a real or complex vector space is called a norm, recall, if the function is positively homogeneous and sub-additive, and if it vanishes only on the zero vector. The norm formalises the intuition of the length of a vector, familiar to all school students. The ground structure of mathematical analysis is the Banach space, a real or complex vector space with a complete norm, its most classical instances including the Lebesgue spaces of measurable functions and the Sobolev spaces of differentiable functions. In engineering, functions of one real variable, and their discrete analogues, are often called signals, and the classification of signals is a common task. When using norms for that purpose, one first ought to examine how well families of norms separate signals. The family of all norms on a complex vector space separates vectors up to phase. The Lebesgue norms separate functions up to their probability distribution. The Sobolev norms separate signals up to autocorrelation function. Each of these rough assertions summarises a theorem, the statement and proof of which are proposed for the talk. The idea of using norms for classifying signals is quite new. It is natural because the standard norms are intuitive artefacts that have thoroughly been studied, and which, like most analytical notions, are readily applicable to numerical data of any size. But ideas are only as good as they turn out to be in practise.

Keywords: signal, norm, identification, separation, classification