Papers by Hans G . Feichtinger
arXiv (Cornell University), Oct 10, 2018
arXiv (Cornell University), Sep 30, 1997
Fractal and fractal-rate stochastic point processes (FSPPs and FRSPPs) provide useful models for ... more Fractal and fractal-rate stochastic point processes (FSPPs and FRSPPs) provide useful models for describing a broad range of diverse phenomena, including electron transport in amorphous semiconductors, computer-network traffic, and sequences of neuronal action potentials. A particularly useful statistic of these processes is the fractal exponent α, which may be estimated for any FSPP or FRSPP by using a variety of statistical methods. Simulated FSPPs and FRSPPs consistently exhibit bias in this fractal exponent, however, rendering the study and analysis of these processes non-trivial. In this paper, we examine the synthesis and estimation of FRSPPs by carrying out a systematic series of simulations for several different types of FRSPP over a range of design values for α. The discrepancy between the desired and achieved values of α is shown to arise from finite data size and from the character of the
arXiv (Cornell University), Jun 15, 2016
This article describes how the ideas promoted by the fundamental papers published by M. Frazier a... more This article describes how the ideas promoted by the fundamental papers published by M. Frazier and B. Jawerth in the eighties have influenced subsequent developments related to the theory of atomic decompositions and Banach frames for function spaces such as the modulation spaces and Besov-Triebel-Lizorkin spaces. Both of these classes of spaces arise as special cases of two different, general constructions of function spaces: coorbit spaces and decomposition spaces. Coorbit spaces are defined by imposing certain decay conditions on the so-called voice transform of the function/distribution under consideration. As a concrete example, one might think of the wavelet transform, leading to the theory of Besov-Triebel-Lizorkin spaces. Decomposition spaces, on the other hand, are defined using certain decompositions in the Fourier domain. For Besov-Triebel-Lizorkin spaces, one uses a dyadic decomposition, while a uniform decomposition yields modulation spaces. Only recently, the second author has established a fruitful connection between modern variants of wavelet theory with respect to general dilation groups (which can be treated in the context of coorbit theory) and a particular family of decomposition spaces. In this way, optimal inclusion results and invariance properties for a variety of smoothness spaces can be established. We will present an outline of these connections and comment on the basic results arising in this context.
Multimedia und Computeranwendungen in der Lehre, 1992
As a result of a research project on "Experimental signal analysis" we present a collection of mo... more As a result of a research project on "Experimental signal analysis" we present a collection of more than 600 MATLAB M-files, called IRSATOL := the IRregular SAmpling TOoLbox. The key problem concerns the reconstruction of a function of one or two variables from sampling values, which are taken at irregularly spaced sampling positions, and which is known to contain only a certain number of frequencies (given spectrum). There are many different solutions to this problem by now. Among the constructive method an iterative version, the so-called" adaptive weights method:' has turned out to be most efficient. For theoretical background see a series of papers by Feichtinger and Grochenig (d. [FG,FCH,FCS,G]). The emphasis of this contribution, however, is on the system built around this large collection of M-files, in order to keep it well organized, and a variety of utilities which will be useful for many other MATLAB projects, helping to make documentation and production of new M-files more efficient. The toolbox can be used both interactively or automaticaly (tutorial style). As MATLAB itself the package is working in a machine independent way.
arXiv (Cornell University), Apr 13, 2020
In the first part of the paper we provide a survey of recent results concerning the problem of po... more In the first part of the paper we provide a survey of recent results concerning the problem of pointwise convergence of integral kernels in Feynman path integral, obtained by means of time-frequency analysis techniques. We then focus on exceptional times, where the previous results do not hold, and we show that weaker forms of convergence still occur. In conclusion we offer some clues about possible physical interpretation of exceptional times.
Rocky Mountain Journal of Mathematics, 1989
Given s G R and 1 < p, q < oo the modulation space Mp g (R m ) can be described as follows, using... more Given s G R and 1 < p, q < oo the modulation space Mp g (R m ) can be described as follows, using the Gauss-function go,go(x) '•= exp(-x 2 ) for the modulation operator. Among these spaces one has the classical potential spaces £2( R m) _ jv/| 2 (R m ) and the remarkable Segal algebra 5 0 (R m ) = Mf^R™). It is the aim of this paper to show that for these spaces an atomic characterization similar to known characterization of Besov spaces can be given (with dilation being replaced by modulation). Our main theorem is the following: Given s G R and some So # 0, go G Mj*j(R m ) (e.g., g G 5(R m ) or g G L 1 with compactly supported Fourier transform) one has: THEOREM . There exist ao > 0 and ßo > 0 such that, for a < a 0 and ß < ßo, there exists C = C(a,ß) > 0 with the following property: f e M^q(R m ) if and only iff = Yin,k a n,kMß n L ak go, for some double sequence of coefficients satisfying [E (El 0 ».*!")*^1 + Wr] 1/9 < c||/|M p yR m )||. The convergence is in the sense of tempered distributions, and in the norm sense for p,q < oo.
arXiv (Cornell University), May 13, 2011
A notion of band limited functions is considered in the case of the hyperbolic plane in its Poinc... more A notion of band limited functions is considered in the case of the hyperbolic plane in its Poincare upper half-plane H realization. The concept of band-limitedness is based on the existence of the Helgason-Fourier transform on H. An iterative algorithm is presented, which allows to reconstruct bandlimited functions from some countable sets of their values. It is shown that for sufficiently dense metric lattices a geometric rate of convergence can be guaranteed as long as the sampling density is high enough compared to the band-width of the sampled function.
arXiv (Cornell University), Mar 8, 2007
Gabor frames for signals over finite Abelian groups, generated by an arbitrary lattice within the... more Gabor frames for signals over finite Abelian groups, generated by an arbitrary lattice within the finite time-frequency plane, are the central topic of this paper. Our generic approach covers both multi-dimensional signals as well as non-separable lattices, and in fact the multi-window case as well. Our generic approach includes most of the fundamental facts about Gabor expansions of finite signals for the case of product lattices, as they have been given by Qiu, Wexler-Raz or Tolimieri-Orr, Bastiaans and Van-Leest and others. In our presentation the spreading representation of linear operators between finite-dimensional Hilbert space as well as a symplectic version of Poisson's summation formula over the finite time-frequency plane are essential ingredients. They bring us to the so-called Fundamental Identity of Gabor Analysis. In addition, we highlight projective representations of the time-frequency plane and its subgroups and explain the natural connection to twisted group algebras. In the finite-dimensional setting discussed in this paper these twisted group algebras are just matrix algebras and their structure provides the algebraic framework for the study of the deeper properties of finite-dimensional Gabor frames, independent of the structure theory theorem for finite Abelian groups.
arXiv (Cornell University), 2008
In this paper, we consider the trace theorem for modulation spaces M s p,q , α-modulation spaces ... more In this paper, we consider the trace theorem for modulation spaces M s p,q , α-modulation spaces M s,α p,q and Besov spaces B s p,q. For the modulation space, we obtain the sharp results.
arXiv (Cornell University), Nov 28, 2008
In this paper, we consider the trace theorem for modulation spaces M s p,q , α-modulation spaces ... more In this paper, we consider the trace theorem for modulation spaces M s p,q , α-modulation spaces M s,α p,q and Besov spaces B s p,q. For the modulation space, we obtain the sharp results.
Lecture Notes in Mathematics, 2008
The use of general descriptive names, registered names, trademarks, etc. in this publication does... more The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
arXiv (Cornell University), Jan 22, 2015
We consider Gabor localization operators G φ,Ω defined by two parameters, the generating function... more We consider Gabor localization operators G φ,Ω defined by two parameters, the generating function φ of a tight Gabor frame {φ λ } λ∈Λ , parametrized by the elements of a given lattice Λ ⊂ R 2 , i.e. a discrete cocompact subgroup of R 2 , and a lattice localization domain Ω ⊂ R 2 with its boundary consisting of line segments connecting points of Λ. We find an explicit formula for the boundary form BF (φ, Ω) = A Λ lim R→∞
arXiv (Cornell University), Apr 3, 2019
In this paper we develop a new way to study the global existence and uniqueness for the Navier-St... more In this paper we develop a new way to study the global existence and uniqueness for the Navier-Stokes equation (NS) and consider the initial data in a class of modulation spaces E s p,q with exponentially decaying weights (s < 0, 1 < p, q < ∞) for which the norms are defined by f E s p,q = k∈Z d 2 s|k|q F −1 χ k+[0,1] d F f q p 1/q. The space E s p,q is a rather rough function space and cannot be treated as a subspace of tempered distributions. For example, we have the embedding H σ ⊂ E s 2,1 for all σ < 0 and s < 0. It is known that H σ (σ < d/2 − 1) is a super-critical space of NS, it follows that E s 2,1 (s < 0) is also super-critical for NS. We show that NS has a unique global mild solution if the initial data belong to E s 2,1 (s < 0) and their Fourier transforms are supported in R d I := {ξ ∈ R d : ξ i 0, i = 1, ..., d}. Similar results hold for the initial data in E s r,1 with 2 < r d. Our results imply that NS has a unique global solution if the initial value u 0 is in L 2 with supp u 0 ⊂ R d I .
Pseudo-Differential Operators, 2008
Our aim is to show how semi-classical analysis can be useful in questions of stability appearing ... more Our aim is to show how semi-classical analysis can be useful in questions of stability appearing in hydrodynamics. We will emphasize on the motivating examples and see how these problems can be solved or by harmonic approximation techniques used in the semi-classical analysis of the Schrödinger operator or by recently obtained semi-classical versions of estimates for operators of principal type (mainly subelliptic estimates). These notes correspond to an extended version of the course given at the school in Cetraro. We have in particularly kept the structure of these lectures with an alternance between the motivating examples and the presentation of the theory. Many of the results which are presented have been obtained in collaboration with Olivier Lafitte.
1996 IEEE International Symposium on Circuits and Systems. Circuits and Systems Connecting the World. ISCAS 96
For the sake of brevity, we shall henceforth use the term filter bank (FB) instead of uniform fil... more For the sake of brevity, we shall henceforth use the term filter bank (FB) instead of uniform filter bank. 2We note that our theory can easily be extended to PR with nonzero delay.
2004 12th European Signal Processing Conference, 2004
We extend a method described by Sintes/Schultz (1998) for the removal of coherent signals in stat... more We extend a method described by Sintes/Schultz (1998) for the removal of coherent signals in stationary broadband noise to the case of non-stationary broadband noise by applying time-frequency methods. The new method is applied to the removal of cardiopulmonary resuscitation (CPR) artifacts in the ECG of domestic pig suffering from ventricular fibrillation (VF). The excellent filtering properties and the possibility of real-time signal processing might have applications in emergency medical settings.
Indagationes Mathematicae (Proceedings), 1974
SPIE Proceedings, 1995
Recently connections between the wavelet transform and filter banks have been established. We sho... more Recently connections between the wavelet transform and filter banks have been established. We show that similar relations exist between the Gabor expansion and DFT filter banks. We introduce the "z-Zak transform" by suitably extending the discrete-time Zak transform and show its equivalence to the polyphase representation. A systematic discussion of parallels between DFT filter banks and Weyl-Heisenberg frames (Gabor expansion theory) is then given. Among other results, it is shown that tight Weyl-Heisenberg frames correspond to paraunitary DFT filter banks. important linear time-frequency representation known as the Gabor expansion79 and the computationally efficient DFT filter banks'°'4'6 can be unified in a similar manner.15 We show that the theory of Weyl-Heisenberg frames (WHFs) ,16-18 which is a fundamental concept in Gabor expansion theory, allows to establish known and new results on DFT filter banks. We also extend the Zak transfoin,'92' a transformation particularly useful for the Gabor expansion, to the complex plane (z-plane), and we show that the resulting "z-Zak transform" is equivalent to the polyphase representation used in filter bank theory.22"°'6"2"3 This paper is organized as follows. Section 1 gives a brief review of the discrete-time Gabor expansion and DFT filter banks, along with the associated concepts of WHFs, Zak transform, and polyphase representation. In Section 2, we introduce the z-Zak transform, show its equivalence to the polyphase representation, and consider its application to the Gabor expansion. Extending previous results,23 we show in Section 3 that the discrete-time Gabor expansion can be interpreted as a DFT filter bank with perfect reconstruction, and we present a systematic discussion of the parallels existing between WHFs (Gabor expansion theory) and DFT filter banks. Our main results can be summarized as follows:
2010 IEEE 11th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC), 2010
We propose advanced compressive estimators of doubly dispersive channels within multicarrier comm... more We propose advanced compressive estimators of doubly dispersive channels within multicarrier communication systems (including classical OFDM systems). The performance of compressive channel estimation has been shown to be limited by leakage components impairing the channel's effective delay-Doppler sparsity. We demonstrate a group sparse structure of these leakage components and apply recently proposed recovery techniques for group sparse signals. We also present a basis optimization method for enhancing group sparsity. Statistical knowledge about the channel can be incorporated in the basis optimization if available. The proposed estimators outperform existing compressive estimators with respect to estimation accuracy and, in one instance, also computational complexity.
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Papers by Hans G . Feichtinger