Friedrich Philipp

Introduction to Finite Frame Theory

To date, frames have established themselves as a standard notion in applied mathematics, computer science, and engineering as a means to derive redundant, yet stable decompositions of a signal for analysis or transmission, while also promoting sparse expansions. The reconstruction procedure is then based on one of the associated dual frames, which—in the case of a Parseval frame—can be chosen to be the frame itself. In this chapter, we provide a comprehensive review of the basics of finite frame theory upon which the subsequent chapters are based. After recalling some background information on Hilbert space theory and operator theory, we introduce the notion of a frame along with some crucial properties and construction procedures. Then we discuss algorithmic aspects such as basic reconstruction algorithms and present brief introductions to diverse applications and extensions of frames. The subsequent chapters of this book will then extend key topics in many intriguing directions.

Measures of Scalability

Scalable frames are frames with the property that the frame vectors can be rescaled resulting in tight frames. However, if a frame is not scalable, one has to aim for an approximate procedure. For this, in this paper we introduce three novel quantitative measures of the closeness to scalability for frames in finite dimensional real Euclidean spaces. Besides the natural measure of scalability given by the distance of a frame to the set of scalable frames, another measure is obtained by optimizing a quadratic functional, while the third is given by the volume of the ellipsoid of minimal volume containing the symmetrized frame. After proving that these measures are equivalent in a certain sense, we establish bounds on the probability of a randomly selected frame to be scalable. In the process, we also derive new necessary and sufficient conditions for a frame to be scalable.

Estimates on the non-real eigenvalues of regular indefinite Sturm–Liouville problems

Regular Sturm-Liouville problems with indefinite weight functions may possess finitely many non-real eigenvalues. In this note we prove explicit bounds on the real and imaginary parts of these eigenvalues in terms of the coefficients of the differential expression.

On Domains of Powers of Linear Operators and Finite Rank Perturbations

Let S and T be linear operators in a linear space such that S ⊂ T. In this note an estimate for the codimension of dom S n in dom T n in terms of the codimension of dom S in dom T is obtained. An immediate consequence is that for any polynomial p the operator p(S) is a finite-dimensional restriction of the operator p(T) whenever S is a finite-dimensional restriction of T. The general results are applied to a perturbation problem of self-adjoint definitizable operators in Krein spaces.

G-Self-adjoint Operators in Almost Pontryagin Spaces

An Almost Pontryagin space \( \left( {\mathcal{H},[ \cdot , \cdot ]} \right) \) admits a decomposition

Signal recovery from thresholded frame measurements

\mathcal{H} = \mathcal{H}_ + [\dot + ]\mathcal{H}_ - [\dot + ]\mathcal{H}^\circ ,

A Balian–Low Theorem for Subspaces

where \( \left( {\mathcal{H}_ + ,[ \cdot , \cdot ]} \right) \) and \( \left( {\mathcal{H}_ - , - [ \cdot , \cdot ]} \right) \) are Hilbert spaces and \( \mathcal{H}_ - \) — as well as \( \mathcal{H}^\circ \) are finite dimensional. Based on the theory of linear relations we introduce the notion of G-self-adjoint operators in Almost Pontryagin spaces and study their spectral properties. In particular, we construct a spectral function for G-self-adjoint operators in Almost Pontryagin spaces. Finally, we apply our results to the Klein-Gordon equation.

Finite sensor Dynamical Sampling

The recently introduced and characterized scalable frames can be considered as those frames which allow for perfect preconditioning in the sense that the frame vectors can be rescaled to yield a tight frame. In this paper we define m−scalability, a refinement of scalability based on the number of non-zero weights used in the rescaling process. We enlighten a close connection between this notion and elements from convex geometry. Another focus lies in the topology of scalable frames. In particular, we prove that the set of scalable frames with “usual” redundancy is nowhere dense in the set of frames.