arXiv subjects

Götz Pfander

Publications and source records attributed to Götz Pfander.

4 recordsLinked to original sources

arXiv2026

Schwartz-Class Gabor Windows and the Balian-Low Classification

Lattice Gabor frames consist of a window function and its time--frequency shifts by the elements of a lattice. This paper addresses the decay and smoothness achievable by a Gabor-frame window for a prescribed time--frequency/phase-space lattice, and gives a classification apart from the borderline cases specified below. Central to the paper is the introduction of an integer lattice parameter, the symplectic index gap. It governs the availability of regular, well-localized windows for a given lattice. For example, we show that a Schwartz-class Gabor-frame window exists exactly when this gap is at least the signal-space dimension, with the extended gap convention for symplectically irrational lattices. We formulate symplectic index gap Balian--Low theorems that place the classical and amalgam Balian--Low theorems in a common framework using the scale of modulation spaces. For symplectically rational lattices---which include those spanned by vectors with rational entries and which are the ones most relevant in applications---the necessary conditions for modulation-space regularity are also sufficient, except at the remaining lower-gap infinity endpoints. The companion paper "Tilings, Packings, and Smooth and Compactly Supported Gabor Windows" proves the associated geometric Tiling--$\varepsilon$--Packing theorem and constructs smooth, compactly supported Gabor windows.

math.FA↗

arXiv2026

Tilings, packings, and the existence of Schwartz-class Gabor windows

The existence and construction of Schwartz-class windows for lattice Gabor frames is a central problem in time--frequency analysis. For general time--frequency lattices, we characterize exactly those lattices that admit Schwartz-class windows and, additionally, windows in the Feichtinger algebra. Thereby we arrive at sharp classical and amalgam Balian--Low theorems for lattices in $\mathbb{R}^d$. For separable lattices and specified block-zero forms, we construct smooth windows that are compactly supported either in time or in frequency. These constructions are based on a characterization of pairs of lattices for which there exists a single set that tiles with one lattice and whose $\varepsilon$-neighborhood packs with the other.

math.FA↗

arXiv2013

Local sampling and approximation of operators with bandlimited Kohn-Nirenberg symbols

Recent sampling theorems allow for the recovery of operators with bandlimited Kohn-Nirenberg symbols from their response to a single discretely supported identifier signal. The available results are inherently non-local. For example, we show that in order to recover a bandlimited operator precisely, the identifier cannot decay in time nor in frequency. Moreover, a concept of local and discrete representation is missing from the theory. In this paper, we develop tools that address these shortcomings. We show that to obtain a local approximation of an operator, it is sufficient to test the operator on a truncated and mollified delta train, that is, on a compactly supported Schwarz class function. To compute the operator numerically, discrete measurements can be obtained from the response function which are localized in the sense that a local selection of the values yields a local approximation of the operator. Central to our analysis is to conceptualize the meaning of localization for operators with bandlimited Kohn-Nirenberg symbol.

math.FA↗

arXiv2011

Reconstruction and Estimation of Scattering Functions of Overspread Radar Targets

In many radar scenarios, the radar target or the medium is assumed to possess randomly varying parts. The properties of a target are described by a random process known as the spreading function. Its second order statistics under the WSSUS assumption are given by the scattering function. Recent developments in the operator identification theory suggest a channel sounding procedure that allows to determine the spreading function given complete statistical knowledge of the operator echo. We show that in a continuous model it is indeed theoretically possible to identify a scattering function of an overspread target given full statistics of a received echo from a single sounding by a custom weighted delta train. Our results apply whenever the scattering function is supported on a set of area less than one. Absent such complete statistics, we construct and analyze an estimator that can be used as a replacement of the averaged periodogram estimator in case of poor geometry of the support set of the scattering function.

cs.IT↗