## Fachbereich Mathematik

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- 1995 (39) (remove)

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#### Keywords

- Boltzmann Equation (3)
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- Evolution Equations (1)
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- Non-linear wavelet thresholding (1)
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- Wavelet Thresholding: Beyond the Gaussian I.I.D. Situation (1995)
- With this article we first like to give a brief review on wavelet thresholding methods in non-Gaussian and non-i.i.d. situations, respectively. Many of these applications are based on Gaussian approximations of the empirical coefficients. For regression and density estimation with independent observations, we establish joint asymptotic normality of the empirical coefficients by means of strong approximations. Then we describe how one can prove asymptotic normality under mixing conditions on the observations by cumulant techniques.; In the second part, we apply these non-linear adaptive shrinking schemes to spectral estimation problems for both a stationary and a non-stationary time series setup. For the latter one, in a model of Dahlhaus on the evolutionary spectrum of a locally stationary time series, we present two different approaches. Moreover, we show that in classes of anisotropic function spaces an appropriately chosen wavelet basis automatically adapts to possibly different degrees of regularity for the different directions. The resulting fully-adaptive spectral estimator attains the rate that is optimal in the idealized Gaussian white noise model up to a logarithmic factor.

- Topologie II (1995)

- The Solution of Linear Inverse Problems in Satellite Geodesy by Means of Spherical Spline Approximation (1995)
- In this paper we consider a certain class of geodetic linear inverse problems LambdaF=G in a reproducing kernel Hilbert space setting to obtain a bounded generalized inverse operator Lambda. For a numerical realization we assume G to be given at a finite number of discrete points to which we employ a spherical spline interpolation method adapted to the Hilbertspaces. By applying Lambda to the obtained spline interpolant we get an approximation of the solution F. Finally our main task is to show some properties of the approximated solution and to prove convergence results if the data set increases.

- Symmetry properties of average densities and tangent measure distributions of measures on the line (1995)
- Answering a question by Bedford and Fisher we show that for every Radon measure on the line with positive and finite lower and upper densities the one-sided average densities always agree with one half of the circular average densities at almost every point. We infer this result from a more general formula, which involves the notion of a tangent measure distribution introduced by Bandt and Graf. This formula shows that the tangent measure distributions are Palm distributions and define self-similar random measures in the sense of U. Zähle.

- Stochastic Reconstruction of Loading Histories from a Rainflow Matrix (1995)
- This paper is devoted to the mathematica l description of the solution of the so-called rainflow reconstruction problem, i.e. the problem of constructing a time series with an a priori given rainflow m atrix. The algorithm we present is mathematically exact in the sense that no app roximations or heuristics are involved. Furthermore it generates a uniform distr ibution of all possible reconstructions and thus an optimal randomization of the reconstructed series. The algorithm is a genuine on-line scheme. It is easy adj ustable to all variants of rainflow such as sysmmetric and asymmetric versions a nd different residue techniques.