Irregular Spline - Type Spaces and Almost Periodic Functions

Frames of regular translates appear in the study of shift invariant spaces and their spanning properties are analyzed using Fourier analysis of periodic functions. Introducing irregular shifts gives rise to an interesting and useful generalization of these frames. They arise, for example, in nonuniform sampling problems or minimal norm interpolation in weighted Sobolev spaces, however they can not be analyzed by imitation of methods of the regular case. The aim of this project is to provide innovative tools for studying irregular systems of translates by investigating connections with almost periodic functions, exploiting convolution results for Wiener amalgam spaces and localization properties of generating families. As Wiener amalgam spaces are ubiquitous in the study of spline type spaces, almost periodic functions, although appearing naturally in the context of irregular frames of translates, have gained little attention in that area. The first objective of the project is to derive sound connections of almost periodic functions to families of irregular shifts and irregular Gabor systems. New mathematical objects, such as Gramian function and Gramian matrix for irregular systems of translates will be introduced. Their properties, studied through relations between translation numbers and Bohr spectrum of almost periodic functions, will be used to derive conditions on the generating window and the set of translates to form a frame or a Riesz basis. For irregular Gabor families we will introduce a notion of Walnut like representations and study frame operators in this new setting. Obtained results will enrich theory of almost periodic function and branches of mathematics where they form a relevant class of functions to study pseudodifferential operators, systems theory or quasicrystals. The second main topic of the project, closely connected with the first one, is the investigation and construction of approximate dual systems. Frame operators associated to irregular frames of translates poses very little structure that can be exploited to invert them efficiently. Moreover, dual systems obtained by applying inverse frame operator to each atom lose their original form and are not systems of translates anymore. Using localization theory and finite section methods applied to Gramian matrix of irregular frame of translates we will construct approximate dual systems that result in good reconstruction errors. Using a concept of gap between subspaces we will quantify resulting reconstruction errors. The third stage of the project is the application of the developed theory to nonuniform sampling in irregular spline type spaces generated by irregular Riesz bases. The computation of an approximate dual Riesz basis through finite section methods will allow for approximate reconstructions and will be an alternative to iterative reconstruction algorithms used so far. The three main topics are connected by common ideas and technical tools as well as the motivation by concrete applications. The achieved results will therefore - partly with partners from the applied sciences - be realized in the context of the project and with a focus on real-life applications.

One of the cornerstones of digital signal analysis is the so-called sampling theorem, according to which a bandlimited signal, or more generally a signal that is an element of some shift invariant space, can be completely reconstructed from the sampling values taken at any set of sufficiently dense regular points. In recent years there has been a lot of interest in nonuniform sampling motivated by real life situations where regular samples may not be possible to obtain, such as in communication theory, medical imaging or astronomical measurements. Sampling problems are approached best through theory of frames. Frames are collections of functions such that any other function can be written as their linear combination. A particular type of frames are frames of translates, and in particular Gabor frames. These are collections of one or more functions that are translated (shifted) by some discrete regular or irregular set. Frames of regular translates arise in the study of shift invariant spaces and their spanning properties are analyzed using Fourier analysis of periodic functions. Introducing irregular shifts gives rise to an interesting and useful generalization of those systems that arise naturally in nonuniform sampling problems. However irregular frames of translates can not be analyzed by imitation of the methods of the regular case. In this project the theory of irregular frames was developed for model sets. Model sets are sets of points that come from certain orthogonal projections of lattice points in higher dimensions. The most important result of the project is the characterization of the frame operator associated to frames of translates for model sets, or Gabor frames for model sets, respectively. For Gabor frames on lattices, a representation of the Gabor frame operator is known in the literature as Janssen representation. Such a characterization in terms of almost periodic functions and their Fourier series expansion allows to formulate conditions on the pair of families to be dual to each other. For many model sets it is not possible to find pairs of dual systems that are both frames of translates or Gabor frames, respectively. In such situations, we can find approximate dual frames of translates, or Gabor frames, and reconstruct a function to a desired accuracy. Such pairs of systems allow for analysis and synthesis of functions from generalised samples, that is inner products of a signal with members of one of the generating systems. As part of the developed theory, we also provide approximation errors.