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Friedrich Philipp

Publications and source records attributed to Friedrich Philipp.

At least 37 records · Page 2Linked to original sources
arXiv2021

A note on the invertibility of the Gabor frame operator on certain modulation spaces

We consider Gabor frames generated by a general lattice and a window function that belongs to one of the following spaces: the Sobolev space $V_1 = H^1(\mathbb R^d)$, the weighted $L^2$-space $V_2 = L_{1 + |x|}^2(\mathbb R^d)$, and the space $V_3 = \mathbb H^1(\mathbb R^d) = V_1 \cap V_2$ consisting of all functions with finite uncertainty product; all these spaces can be described as modulation spaces with respect to suitable weighted $L^2$ spaces. In all cases, we prove that the space of Bessel vectors in $V_j$ is mapped bijectively onto itself by the Gabor frame operator. As a consequence, if the window function belongs to one of the three spaces, then the canonical dual window also belongs to the same space. In fact, the result not only applies to frames, but also to frame sequences.

math.FA↗
arXiv2021

A quantitative subspace Balian-Low theorem

Let $\mathcal G\subset L^2(\mathbb R)$ be the subspace spanned by a Gabor Riesz sequence $(g,Λ)$ with $g\in L^2(\mathbb R)$ and a lattice $Λ\subset\mathbb R^2$ of rational density. It was shown recently that if $g$ is well-localized both in time and frequency, then $\mathcal G$ cannot contain any time-frequency shift $π(z) g$ of $g$ with $z\notinΛ$. In this paper, we improve the result to the quantitative statement that the $L^2$-distance of $π(z)g$ to the space $\mathcal G$ is equivalent to the Euclidean distance of $z$ to the lattice $Λ$, in the sense that the ratio between those two distances is uniformly bounded above and below by positive constants. On the way, we prove several results of independent interest, one of them being closely related to the so-called weak Balian-Low theorem for subspaces.

math.FA↗
arXiv2021

Control of port-Hamiltonian systems with minimal energy supply

We investigate optimal control of linear port-Hamiltonian systems with control constraints, in which one aims to perform a state transition with minimal energy supply. Decomposing the state space into dissipative and non-dissipative (i.e. conservative) subspaces, we show that the set of reachable states is bounded w.r.t. the dissipative subspace. We prove that the optimal control problem exhibits the turnpike property with respect to the non-dissipative subspace, i.e., for varying initial conditions and time horizons optimal state trajectories evolve close to the conservative subspace most of the time. We analyze the corresponding steady-state optimization problem and prove that all optimal steady states lie in the non-dissipative subspace. We conclude this paper by illustrating these results by a numerical example from mechanics.

math.OC↗
arXiv2020

Finite Rank Perturbations of Linear Relations and Matrix Pencils

We elaborate on the deviation of the Jordan structures of two linear relations that are finite-dimensional perturbations of each other. We compare their number of Jordan chains of length at least $n$. In the operator case, it was recently proved that the difference of these numbers is independent of $n$ and is at most the defect between the operators. One of the main results of this paper shows that in the case of linear relations this number has to be multiplied by $n+1$ and that this bound is sharp. The reason for this behavior is the existence of singular chains. We apply our results to one-dimensional perturbations of singular and regular matrix pencils. This is done by representing matrix pencils via linear relations. This technique allows for both proving known results for regular pencils as well as new results for singular ones.

math.FA↗
arXiv2020

Spectral enclosures for a class of block operator matrices

We prove new spectral enclosures for the non-real spectrum of a class of $2\times2$ block operator matrices with self-adjoint operators $A$ and $D$ on the diagonal and operators $B$ and $-B^*$ as off-diagonal entries. One of our main results resembles Gershgorin's circle theorem. The enclosures are applied to $J$-frame operators.

math.SP↗
arXiv2018

Frame Properties of Operator Orbits

We consider sequences in a Hilbert space $\mathcal H$ of the form $(T^nf_0)_{n\in I},$ with a linear operator $T$, the index set being either $I = \mathbb N$ or $I = \mathbb Z$, a vector $f_0\in \mathcal H$, and answer the following two related questions: (a) {\it Which frames for $\mathcal H$ are of this form with an at least closable operator $T$?} and (b) {\it For which bounded operators $T$ and vectors $f_0$ is $(T^nf_0)_{n\in I}$ a frame for $\mathcal H$?} As a consequence of our results, it turns out that an overcomplete Gabor or wavelet frame can never be written in the form $(T^nf_0)_{n\in\mathbb N}$ with a bounded operator $T$. The corresponding problem for $I = \mathbb Z$ remains open. Despite the negative result for Gabor and wavelet frames, the results demonstrate that the class of frames that can be represented in the form $(T^nf_0)_{n\in\mathbb N}$ with a bounded operator $T$ is significantly larger than what could be expected from the examples known so far.

math.FA↗
arXiv2016

Bessel orbits of normal operators

Given a bounded normal operator $A$ in a Hilbert space and a fixed vector $x$, we elaborate on the problem of finding necessary and sufficient conditions under which $(A^kx)_{k\in\mathbb N}$ constitutes a Bessel sequence. We provide a characterization in terms of the measure $\|E(\cdot)x\|^2$, where $E$ is the spectral measure of the operator $A$. In the separately treated special cases where $A$ is unitary or selfadjoint we obtain more explicit characterizations. Finally, we apply our results to a sequence $(A^kx)_{k\in\mathbb N}$, where $A$ arises from the heat equation. The problem is motivated by and related to the new field of Dynamical Sampling which was recently initiated by Aldroubi et al.

math.FA↗
arXiv2016

The Effect of Perturbations of Operator-Valued Frame Sequences and Fusion Frames on Their Duals

Fusion frames, and, more generally, operator-valued frame sequences are generalizations of classical frames, which are today a standard notion when redundant, yet stable sequences are required. However, the question of stability of duals with respect to perturbations has not been satisfactorily answered. In this paper, we quantitatively measure this stability by considering the associated deviations of the canonical and alternate dual sequences from the original ones. It is proven that operator-valued frame sequences are indeed stable in this sense. Along the way, we also generalize existing definitions for fusion frame duals to the infinite-dimensional situation and analyze how they perform with respect to a list of desiderata which, to our minds, a fusion frame dual should satisfy. Finally, we prove a similar stability result as above for fusion frames and their canonical duals.

math.FA↗
arXiv2016

The effect of perturbations of linear operators on their polar decomposition

The effect of matrix perturbations on the polar decomposition has been studied by several authors and various results are known. However, for operators between infinite-dimensional spaces the problem has not been considered so far. Here, we prove in particular that the partial isometry in the polar decomposition of an operator is stable under perturbations, given that kernel and range of original and perturbed operator satisfy a certain condition. In the matrix case, this condition is weaker than the usually imposed equal-rank condition. It includes the case of semi-Fredholm operators with agreeing nullities and deficiencies, respectively. In addition, we prove a similar perturbation result where the ranges or the kernels of the two operators are assumed to be sufficiently close to each other in the gap metric.

math.FA↗
arXiv2014

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.

math.FA↗
arXiv2014

Spectral Points of Type $π_+$ and Type $π_-$ of Closed Operators in Indefinite Inner Product Spaces

We introduce the notion of spectral points of type $π_+$ and type $π_-$ of closed operators $A$ in a Hilbert space which is equipped with an indefinite inner product. It is shown that these points are stable under compact perturbations. In the second part of the paper we assume that $A$ is symmetric with respect to the indefinite inner product and prove that the growth of the resolvent of $A$ is of finite order in a neighborhood of a real spectral point of type $π_+$ or $π_-$ which is not in the interior of the spectrum of $A$. Finally, we prove that there exists a local spectral function on intervals of type $π_+$ or $π_-$.

math.SP↗
arXiv2014

Phase Retrieval from 4N-4 Measurements

We prove by means of elementary methods that phase retrieval of complex polynomials p of degree less than N is possible with 4N-4 phaseless Fourier measurements of p and p'. In addition, we provide an associated algorithm and prove that it recovers p up to global phase.

math.FA↗
arXiv2014

Scalable Frames and Convex Geometry

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, and study the connection between this notion and elements from convex geometry. Finally, we provide results on the topology of scalable frames. In particular, we prove that the set of scalable frames with "small" redundancy is nowhere dense in the set of frames.

math.NA↗
arXiv2012

The numerical range of non-negative operators in Krein spaces

We define and characterize the Krein space numerical range $W(A)$ and the Krein space co-numerical range $W_{\rm co}(A)$ of a non-negative operator $A$ in a Krein space. It is shown that the non-zero spectrum of $A$ is contained in the closure of $W(A)\cap W_{\rm co}(A)$.

math.FA↗
arXiv2012

Indefinite Sturm-Liouville operators with periodic coefficients

We investigate the spectral properties of the maximal operator $A$ associated with a differential expression $\frac 1 w(-\frac d {dx}(p\frac d {dx}) + q)$ with real-valued periodic coefficients $w$, $p$ and $q$ where $w$ changes sign. It turns out that the non-real spectrum of $A$ is bounded, symmetric with respect to the real axis and consists of a finite number of analytic curves. The real spectrum is band-shaped and neither bounded from above nor from below. We characterize the finite spectral singularities of $A$ and prove that there is only a finite number of them. Finally, we provide a condition on the coefficients which ensures that $\infty$ is not a spectral singularity of $A$.

math.SP↗
arXiv2012

Scalable Frames

Tight frames can be characterized as those frames which possess optimal numerical stability properties. In this paper, we consider the question of modifying a general frame to generate a tight frame by rescaling its frame vectors; a process which can also be regarded as perfect preconditioning of a frame by a diagonal operator. A frame is called scalable, if such a diagonal operator exists. We derive various characterizations of scalable frames, thereby including the infinite-dimensional situation. Finally, we provide a geometric interpretation of scalability in terms of conical surfaces.

math.NA↗