KLUEDO RSS FeedKLUEDO Dokumente/documents
https://kluedo.ub.uni-kl.de/index/index/
Tue, 08 Feb 2011 12:16:54 +0100Tue, 08 Feb 2011 12:16:54 +0100Locally Supported Wavelets for the Separation of Spherical Vector Fields with Respect to their Sources
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2284
We provide a space domain oriented separation of magnetic fields into parts generated by sources in the exterior and sources in the interior of a given sphere. The separation itself is well-known in geomagnetic modeling, usually in terms of a spherical harmonic analysis or a wavelet analysis that is spherical harmonic based. However, it can also be regarded as a modification of the Helmholtz decomposition for which we derive integral representations with explicitly known convolution kernels. Regularizing these singular kernels allows a multiscale representation of the magnetic field with locally supported wavelets. This representation is applied to a set of CHAMP data for crustal field modeling.Christian Gerhardsreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2284Tue, 08 Feb 2011 12:16:54 +0100Limit Formulae and Jump Relations of Potential Theory in Sobolev Spaces
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2132
In this article we combine the modern theory of Sobolev spaces with the classical theory of limit formulae and jump relations of potential theory. Also other authors proved the convergence in Lebesgue spaces for integrable functions. The achievement of this paper is the L2 convergence for the weak derivatives of higher orders. Also the layer functions F are elements of Sobolev spaces and a two dimensional suitable smooth submanifold in R3, called regular Cm-surface. We are considering the potential of the single layer, the potential of the double layer as well as their first order normal derivatives. Main tool is the convergence in Cm-norm which is proved with help of some results taken from [14]. Additionally, we need a result about the limit formulae in L2-norm, which can be found in [16], and a reduction result which we took from [19]. Moreover we prove the convergence in the Hölder spaces Cm,alpha. Finally, we give an application of the limit formulae and jump relations to Geomathematics. We generalize a density results, see e.g. [11], from L2 to Hm,2. For it we prove the limit formula for U1 in (Hm,2)' also.Thomas Raskop; Martin Grothauspreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2132Mon, 21 Sep 2009 20:04:59 +0200Wärmetransportmodellierung in tiefen geothermischen Systemen
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2129
Gegenstand dieser Arbeit ist die Entwicklung eines Wärmetransportmodells für tiefe geothermische (hydrothermale) Reservoire. Existenz- und Eindeutigkeitsaussagen bezüglich einer schwachen Lösung des vorgestellten Modells werden getätigt. Weiterhin wird ein Verfahren zur Approximation dieser Lösung basierend auf einem linearen Galerkin-Schema dargelegt, wobei sowohl die Konvergenz nachgewiesen als auch eine Konvergenzrate erarbeitet werden.Isabel Ostermannreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2129Sat, 19 Sep 2009 16:50:42 +0200Spherical Fast Multiscale Approximation by Locally Compact Orthogonal Wavelets
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2128
Using a stereographical projection to the plane we construct an O(N log(N)) algorithm to approximate scattered data in N points by orthogonal, compactly supported wavelets on the surface of a 2-sphere or a local subset of it. In fact, the sphere is not treated all at once, but is split into subdomains whose results are combined afterwards. After choosing the center of the area of interest the scattered data points are mapped from the sphere to the tangential plane through that point. By combining a k-nearest neighbor search algorithm and the two dimensional fast wavelet transform a fast approximation of the data is computed and mapped back to the sphere. The algorithm is tested with nearly 1 million data points and yields an approximation with 0.35% relative errors in roughly 2 minutes on a standard computer using our MATLAB implementation. The method is very flexible and allows the application of the full range of two dimensional wavelets.Frank Bauer; Martin Guttingpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2128Thu, 17 Sep 2009 18:59:29 +0200The outer oblique boundary problem of potential theory
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2110
In this article we prove existence and uniqueness results for solutions to the outer oblique boundary problem for the Poisson equation under very weak assumptions on boundary, coefficients and inhomogeneities. Main tools are the Kelvin transformation and the solution operator for the regular inner problem, provided in [1]. Moreover we prove regularisation results for the weak solutions of both, the inner and the outer problem. We investigate the non-admissible direction for the oblique vector field, state results with stochastic inhomogeneities and provide a Ritz-Galerkinm approximation. The results are applicable to problems from Geomathematics, see e.g. [2] and [3].Thomas Raskop; Martin Grothauspreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2110Mon, 13 Jul 2009 17:19:06 +0200Klassische Erdschwerefeldbestimmung aus der Sicht moderner Geomathematik
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2040
Gegenstand dieser Arbeit ist die kanonische Verbindung klassischer globaler Schwerefeldmodellierung in der Konzeption von Stokes (1849) und Neumann (1887) und moderner lokaler Multiskalenberechnung mittels lokalkompakter adaptiver Wavelets. Besonderes Anliegen ist die "Zoom-in"-Ermittlung von Geoidhöhen aus lokal gegebenen Schwereanomalien bzw. Schwerestörungen.Willi Freeden; Kerstin Wolfreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2040Tue, 18 Nov 2008 13:50:49 +0100On Mathematical Aspects of a Combined Inversion of Gravity and Normal Mode Variations by a Spline Method
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2038
This paper provides a brief overview of two linear inverse problems concerned with the determination of the Earth’s interior: inverse gravimetry and normal mode tomography. Moreover, a vector spline method is proposed for a combined solution of both problems. This method uses localised basis functions, which are based on reproducing kernels, and is related to approaches which have been successfully applied to the inverse gravimetric problem and the seismic traveltime tomography separately.Paula Berkel; Volker Michelpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2038Mon, 03 Nov 2008 15:10:09 +0100Classical Globally Reflected Gravity Field Determination in Modern Locally Oriented Multiscale Framework
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2031
The purpose of this paper is the canonical connection of classical global gravity field determination following the concept of Stokes (1849), Bruns (1878), and Neumann (1887) on the one hand and modern locally oriented multiscale computation by use of adaptive locally supported wavelets on the other hand. Essential tools are regularization methods of the Green, Neumann, and Stokes integral representations. The multiscale approximation is guaranteed simply as linear difference scheme by use of Green, Neumann, and Stokes wavelets, respectively. As an application, gravity anomalies caused by plumes are investigated for the Hawaiian and Iceland areas.W. Freeden; T. Fehlinger; M. Klug; D. Mathar; K. Wolfreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2031Tue, 14 Oct 2008 12:34:09 +0200Speech Recognition Support of Assisted Living
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2029
We present results and views about a project in assisted living. The scenario is a room in which an elderly and/or disabled person lives who is not able to perform certain actions due to restricted mobility. We enable the person to express commands verbally that will then be executed automatically. There are several severe problems involved that complicate the situation. The person may utter the command in a rather unexpected way, the person makes an error or the action cannot be performed due to several reasons. In our approach we present an architecture with three components: The recognition component that contains novel features in the signal processing, the analysis component that logically analyzes the command, and the execution component that performs the action automatically. All three components communicate with each other.Georg Hebinger; Volker Michel; Michael Richter; Andreas Simonpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/2029Wed, 01 Oct 2008 16:07:44 +0200On the Local Multiscale Determination of the Earth`s Disturbing Potential From Discrete Deflections of the Vertical
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1947
As a first approximation the Earth is a sphere; as a second approximation it may be considered an ellipsoid of revolution. The deviations of the actual Earth's gravity field from the ellipsoidal 'normal' field are so small that they can be understood to be linear. The splitting of the Earth's gravity field into a 'normal' and a remaining small 'disturbing' field considerably simplifies the problem of its determination. Under the assumption of an ellipsoidal Earth model high observational accuracy is achievable only if the deviation (deflection of the vertical) of the physical plumb line, to which measurements refer, from the ellipsoidal normal is not ignored. Hence, the determination of the disturbing potential from known deflections of the vertical is a central problem of physical geodesy. In this paper we propose a new, well-promising method for modelling the disturbing potential locally from the deflections of the vertical. Essential tools are integral formulae on the sphere based on Green's function of the Beltrami operator. The determination of the disturbing potential from deflections of the vertical is formulated as a multiscale procedure involving scale-dependent regularized versions of the surface gradient of the Green function. The modelling process is based on a multiscale framework by use of locally supported surface curl-free vector wavelets.Willi Freeden; Thomas Fehlinger; Carsten Mayer; Michael Schreinerpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1947Mon, 07 Apr 2008 17:08:11 +0200On the Completeness and Closure of Vector and Tensor Spherical Harmonics
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1943
An intrinsically on the 2-sphere formulated proof of the closure and completeness of spherical harmonics is given in vectorial and tensorial framework. The considerations are essentially based on vector and tensor approximation in terms of zonal vector and tensor Bernstein kernels, respectively.Willi Freeden; Martin Guttingpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1943Mon, 07 Apr 2008 13:48:15 +0200Time-Space Multiscale Analysis by Use of Tensor Product Wavelets and its Application to Hydrology and GRACE Data
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1924
This paper presents a wavelet analysis of temporal and spatial variations of the Earth's gravitational potential based on tensor product wavelets. The time--space wavelet concept is realized by combining Legendre wavelets for the time domain and spherical wavelets for the space domain. In consequence, a multiresolution analysis for both, temporal and spatial resolution, is formulated within a unified concept. The method is then numerically realized by using first synthetically generated data and, finally, several real data sets.Helga Nutz; Kerstin Wolfpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1924Fri, 11 Jan 2008 11:11:17 +0100Splines on the 3-dimensional Ball and their Application to Seismic Body Wave Tomography
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1854
In this paper we construct spline functions based on a reproducing kernel Hilbert space to interpolate/approximate the velocity field of earthquake waves inside the Earth based on traveltime data for an inhomogeneous grid of sources (hypocenters) and receivers (seismic stations). Theoretical aspects including error estimates and convergence results as well as numerical results are demonstrated.Abel Amirbekyan; Volker Michelpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1854Tue, 17 Apr 2007 12:51:03 +0200Locally Supported Approximate Identities on the Unit Ball
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1837
We present a constructive theory for locally supported approximate identities on the unit ball in \(\mathbb{R}^3\). The uniform convergence of the convolutions of the derived kernels with an arbitrary continuous function \(f\) to \(f\), i.e. the defining property of an approximate identity, is proved. Moreover, an explicit representation for a class of such kernels is given. The original publication is available at www.springerlink.comMuhammad Akram; Volker Michelpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1837Sun, 11 Feb 2007 15:03:29 +0100Local Modelling of Sea Surface Topography from (Geostrophic) Ocean Flow
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1836
This paper deals with the problem of determining the sea surface topography from geostrophic flow of ocean currents on local domains of the spherical Earth. In mathematical context the problem amounts to the solution of a spherical differential equation relating the surface curl gradient of a scalar field (sea surface topography) to a surface divergence-free vector field(geostrophic ocean flow). At first, a continuous solution theory is presented in the framework of an integral formula involving Green’s function of the spherical Beltrami operator. Different criteria derived from spherical vector analysis are given to investigate uniqueness. Second, for practical applications Green’s function is replaced by a regularized counterpart. The solution is obtained by a convolution of the flow field with a scaled version of the regularized Green function. Calculating locally without boundary correction would lead to errors near the boundary. To avoid these Gibbs phenomenona we additionally consider the boundary integral of the corresponding region on the sphere which occurs in the integral formula of the solution. For reasons of simplicity we discuss a spherical cap first, that means we consider a continuously differentiable (regular) boundary curve. In a second step we concentrate on a more complicated domain with a non continuously differentiable boundary curve, namely a rectangular region. It will turn out that the boundary integral provides a major part for stabilizing and reconstructing the approximation of the solution in our multiscale procedure.Thomas Fehlinger; Willi Freeden; Simone Gramsch; Carsten Mayer; Dominik Michel; Michael Schreinerreporthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1836Sun, 11 Feb 2007 15:03:15 +0100Numerical Aspects of a Spline-Based Multiresolution Recovery of the Harmonic Mass Density out of Gravity Functionals
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1823
We show the numerical applicability of a multiresolution method based on harmonic splines on the 3-dimensional ball which allows the regularized recovery of the harmonic part of the Earth's mass density distribution out of different types of gravity data, e.g. different radial derivatives of the potential, at various positions which need not be located on a common sphere. This approximated harmonic density can be combined with its orthogonal anharmonic complement, e.g. determined out of the splitting function of free oscillations, to an approximation of the whole mass density function. The applicability of the presented tool is demonstrated by several test calculations based on simulated gravity values derived from EGM96. The method yields a multiresolution in the sense that the localization of the constructed spline basis functions can be increased which yields in combination with more data a higher resolution of the resulting spline. Moreover, we show that a locally improved data situation allows a highly resolved recovery in this particular area in combination with a coarse approximation elsewhere which is an essential advantage of this method, e.g. compared to polynomial approximation.Volker Michel; Kerstin Wolfpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1823Wed, 13 Dec 2006 15:19:35 +0100Fast Approximation on the 2-Sphere by Optimally Localized Approximate Identities
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1822
We introduce a method to construct approximate identities on the 2-sphere which have an optimal localization. This approach can be used to accelerate the calculations of approximations on the 2-sphere essentially with a comparably small increase of the error. The localization measure in the optimization problem includes a weight function which can be chosen under some constraints. For each choice of weight function existence and uniqueness of the optimal kernel are proved as well as the generation of an approximate identity in the bandlimited case. Moreover, the optimally localizing approximate identity for a certain weight function is calculated and numerically tested.Volker Michelpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1822Wed, 13 Dec 2006 13:46:52 +0100Biorthogonal Locally Supported Wavelets on the Sphere Based on Zonal Kernel Functions
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1787
This paper presents a method for approximating spherical functions from discrete data of a block-wise grid structure. The essential ingredients of the approach are scaling and wavelet functions within a biorthogonalisation process generated by locally supported zonal kernel functions. In consequence, geophysically and geodetically relevant problems involving rotation-invariant pseudodifferential operators become attackable. A multiresolution analysis is formulated enabling a fast wavelet transform similar to the algorithms known from one-dimensional Euclidean theory.Willi Freeden; Michael Schreinerpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1787Tue, 24 Oct 2006 22:02:39 +0200Time-Dependent Cauchy-Navier Splines and their Application to Seismic Wave Front Propagation
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1760
In this paper a known orthonormal system of time- and space-dependent functions, that were derived out of the Cauchy-Navier equation for elastodynamic phenomena, is used to construct reproducing kernel Hilbert spaces. After choosing one of the spaces the corresponding kernel is used to define a function system that serves as a basis for a spline space. We show that under certain conditions there exists a unique interpolating or approximating, respectively, spline in this space with respect to given samples of an unknown function. The name "spline" here refers to its property of minimising a norm among all interpolating functions. Moreover, a convergence theorem and an error estimate relative to the point grid density are derived. As numerical example we investigate the propagation of seismic waves.Paula Kammann; Volker Michelpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1760Wed, 23 Aug 2006 10:19:04 +0200Contributions of the Geomathematics Group to the GAMM 76th Annual Meeting
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1700
The following three papers present recent developments in nonlinear Galerkin schemes for solving the spherical Navier-Stokes equation, in wavelet theory based on the 3-dimensional ball, and in multiscale solutions of the Poisson equation inside the ball, that have been presented at the 76th GAMM Annual Meeting in Luxemburg. Part A: A Nonlinear Galerkin Scheme Involving Vectorial and Tensorial Spherical Wavelets for Solving the Incompressible Navier-Stokes Equation on the Sphere The spherical Navier-Stokes equation plays a fundamental role in meteorology by modelling meso-scale (stratified) atmospherical flows. This article introduces a wavelet based nonlinear Galerkin method applied to the Navier-Stokes equation on the rotating sphere. In detail, this scheme is implemented by using divergence free vectorial spherical wavelets, and its convergence is proven. To improve numerical efficiency an extension of the spherical panel clustering algorithm to vectorial and tensorial kernels is constructed. This method enables the rapid computation of the wavelet coefficients of the nonlinear advection term. Thereby, we also indicate error estimates. Finally, extensive numerical simulations for the nonlinear interaction of three vortices are presented. Part B: Methods of Resolution for the Poisson Equation on the 3D Ball Within the article at hand, we investigate the Poisson equation solved by an integral operator, originating from an ansatz by Greens functions. This connection between mass distributions and the gravitational force is essential to investigate, especially inside the Earth, where structures and phenomena are not sufficiently known and plumbable. Since the operator stated above does not solve the equation for all square-integrable functions, the solution space will be decomposed by a multiscale analysis in terms of scaling functions. Classical Euclidean wavelet theory appears not to be the appropriate choice. Ansatz functions are chosen to be reflecting the rotational invariance of the ball. In these terms, the operator itself is finally decomposed and replaced by versions more manageable, revealing structural information about itself. Part C: Wavelets on the 3–dimensional Ball In this article wavelets on a ball in R^3 are introduced. Corresponding properties like an approximate identity and decomposition/reconstruction (scale step property) are proved. The advantage of this approach compared to a classical Fourier analysis in orthogonal polynomials is a better localization of the used ansatz functions.M.J. Fengler; D. Michel; V. Michelpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1700Fri, 20 Jan 2006 15:48:01 +0100A Wavelet Approach to Time-Harmonic Maxwells Equations
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1699
By means of the limit and jump relations of classical potential theory with respect to the vectorial Helmholtz equation a wavelet approach is established on a regular surface. The multiscale procedure is constructed in such a way that the emerging scalar, vectorial and tensorial potential kernels act as scaling functions. Corresponding wavelets are defined via a canonical refinement equation. A tree algorithm for fast decomposition of a complex-valued vector field given on a regular surface is developed based on numerical integration rules. By virtue of this tree algorithm, an effcient numerical method for the solution of vectorial Fredholm integral equations on regular surfaces is discussed in more detail. The resulting multiscale formulation is used to solve boundary-value problems for the time harmonic Maxwell's equations corresponding to regular surfaces.Willi Freeden; Carsten Mayerpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1699Tue, 17 Jan 2006 15:54:18 +0100On the oblique boundary problem with a stochastic inhomogeneity
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1690
We analyze the regular oblique boundary problem for the Poisson equation on a C^1-domain with stochastic inhomogeneities. At first we investigate the deterministic problem. Since our assumptions on the inhomogeneities and coefficients are very weak, already in order to formulate the problem we have to work out properties of functions from Sobolev spaces on submanifolds. An further analysis of Sobolev spaces on submanifolds together with the Lax-Milgram lemma enables us to prove an existence and uniqueness result for weak solution to the oblique boundary problem under very weak assumptions on coefficients and inhomogeneities. Then we define the spaces of stochastic functions with help of the tensor product. These spaces enable us to extend the deterministic formulation to the stochastic setting. Under as weak assumptions as in the deterministic case we are able to prove the existence and uniqueness of a stochastic weak solution to the regular oblique boundary problem for the Poisson equation. Our studies are motivated by problems from geodesy and through concrete examples we show the applicability of our results. Finally a Ritz-Galerkin approximation is provided. This can be used to compute the stochastic weak solution numerically.Thomas Raskop; Martin Grothauspreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1690Thu, 22 Dec 2005 15:04:14 +0100Easy Differentiation and Integration of Homogeneous Harmonic Polynomials
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1688
We will give explicit differentiation and integration rules for homogeneous harmonic polynomial polynomials and spherical harmonics in IR^3 with respect to the following differential operators: partial_1, partial_2, partial_3, x_3 partial_2 - x_2 partial_3, x_3 partial_1 - x_1 partial_3, x_2 partial_1 - x_1 partial_2 and x_1 partial_1 + x_2 partial_2 + x_3 partial_3. A numerical application to the problem of determining the geopotential field will be shown.Frank Bauerpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1688Fri, 02 Dec 2005 23:40:42 +0100Wavelet Modelling of Regional and Temporal Variations of the Earth´s Gravitational Potential
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1641
This work is dedicated to the wavelet modelling of regional and temporal variations of the Earth's gravitational potential observed by GRACE. In the first part, all required mathematical tools and methods involving spherical wavelets are introduced. Then we apply our method to monthly GRACE gravity fields. A strong seasonal signal can be identified, which is restricted to areas, where large-scale redistributions of continental water mass are expected. This assumption is analyzed and verified by comparing the time series of regionally obtained wavelet coefficients of the gravitational signal originated from hydrology models and the gravitational potential observed by GRACE. The results are in good agreement to previous studies and illustrate that wavelets are an appropriate tool to investigate regional time-variable effects in the gravitational field.Martin J. Fengler; Willi Freeden; Annika Kohlhaas; Volker Michel; Thomas Peterspreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1641Thu, 02 Jun 2005 14:48:00 +0200The Spherical Bernstein Wavelet
https://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1637
In this work we introduce a new bandlimited spherical wavelet: The Bernstein wavelet. It possesses a couple of interesting properties. To be specific, we are able to construct bandlimited wavelets free of oscillations. The scaling function of this wavelet is investigated with regard to the spherical uncertainty principle, i.e., its localization in the space domain as well as in the momentum domain is calculated and compared to the well-known Shannon scaling function. Surprisingly, they possess the same localization in space although one is highly oscillating whereas the other one shows no oscillatory behavior. Moreover, the Bernstein scaling function turns out to be the first bandlimited scaling function known to the literature whose uncertainty product tends to the minimal value 1.Martin J. Fengler; Willi Freeden; Martin Guttingpreprinthttps://kluedo.ub.uni-kl.de/frontdoor/index/index/docId/1637Fri, 20 May 2005 12:07:21 +0200