• Prof. Dr. Paul L. Butzer, RWTH Aachen
• Prof. Dr. Gerhard Schmeißer, Universität Erlangen-Nürnberg
• Prof. Dr. Rudolf Stens, RWTH Aachen

Basic relations valid for the Bernstein space B_σ² and their extensions to functions from larger spaces in terms of their distances from B_σ²

This is a trilogy of lectures dealing with theorems valid for bandlimited functions and their extensions to larger function spaces.

The first lecture introduces the theorems from several areas of analysis, both for functions belonging to a Bernstein space of bandlimited function, as well as for functions from spaces of non-bandlimited functions, to be treated in detail.

The second lecture presents a new, unified approach to the errors occurring when the results for bandlimited functions are extended to a function f from a larger space in terms of the distance of f from a suitable Bernstein space. The results also cover the difficult situation of derivative- free error estimates.

The third lecture applies this new approach to the theorems of the first lecture but also treats Hilbert transforms as a further new application of the distance approach.

• Short Abstract
• Extended Abstract

DTU, Lyngby, Denmark

Frames and bases in Hilbert spaces

One of the key issues in harmonic analysis is to consider expansions of functions or signals in terms of simple "building blocks" with desirable features. Classically this has been done using orthonormal bases in Hilbert spaces (or Schauder bases in Banach spaces). However, much more flexibility and much more appealing constructions can be obtained using the modern theory for frame decompositions. The lectures will give an overview of the general theory of frames in Hilbert spaces, as well as a detailed discussion of concrete frames in L² (Gabor frames and wavelet frames).

Yamaguchi University, Ube, Japan

Sampling and recovery of sparse signals and its application to image feature extraction

Many natural signals/images have essential components and they are very few when compared to full dimensions of signals/images. Such sparsity is recently exploited in signal processing, in particular sampling theory for signals with finite rate of innovation (FRI signals). This class of signals includes nonuniform pulse sequences, piecewise polynomial, or piecewise sinusoids. In this talk, we first quickly review the sparsity in signals and images and introduce the definition of the FRI signals. After formulating the sampling process for the signals, we review the classical recovery technique for noiseless case based on annihilating filter. In noisy cases, we have to use optimization approach, such as Cadzow's algorithm, which however does not guarantee any optimality. Hence, we introduce a novel approach which approximately obtains the optimal solution in the sense of maximum likelihood estimation. We further present a precise and robust technique to extract image feature based on the sampling theory.

Universidade do Algarve, Portugal

Operators of Harmonic Analysis in Some Non-Standard Function Spaces

Last decade there was observed a large increase of interest to studies of various operators of harmonic operators, such as maximal, singular, potential, Hardy operators and others, and function spaces in the "variable exponent setting". The latter means thattheparameters defining the operator and/or the space (which usually are constant), may vary from point to point.

In these lectureswe presentbasics of the rapidly developing Variable Exponent Analysis and overviewseveral trends and results in this topic, with the main emphasis on the difficulties arising in the case of variable parameters of operators or spaces.

DePaul University, Chicago, IL, U.S.A.

Sampling and the Energy Concentration Problem: A New Perspective

The Whittaker-Shannon-Kotel’nikov (WSK) sampling theorem plays an important role in communication engineering because it enables engineers to reconstruct bandlimited signals from their samples at a discrete set of points.

Recently, G. Walter and X. Shen introduced another sampling theorem for bandlimited signals in one variable using the prolate spheroidal wave functions (PSWFs), which are eigenfunctions of certain integral and differential operators. They also appear in the solution of the energy concentration problem, which is the problem of finding a bandlimited signal with maximum energy concentration in the interval (−T,T) in the time domain.

In this talk we present a new perspective on sampling and the energy concentration problem by introducing a new generalization of PSWFs.

• Abstract including formulas