• Wolfgang Erb
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• ## Research areas of Dr. Wolfgang Erb

### Uncertainty principles on manifolds

During my time as PhD student in Munich, I was mainly interested in uncertainty principles on manifolds. I was able to show that the product of a space and a frequency variance for functions on a Riemannian manifold is always larger than a particular constant. The key ingredient for the proof of this uncertainty principle is a commutator relation for particular Dunkl operators. In particular, these uncertainties turn out to be sharp. For symmetric spaces, the space variance of the uncertainty product can be used to construct optimally space localized polynomials.

Fig. 1: Two illustrations of optimally space localized polynomials on the unit sphere in different spaces of spherical harmonics.
• Erb, W. Uncertainty principles on compact Riemannian manifolds Appl. Comput. Harmon. Anal. 29, 2 (2010) 182-197
• Erb, W. Uncertainty principles on Riemannian manifolds Logos Verlag Berlin (2010) Dissertation TU München
• Erb, W. Optimally space localized polynomials with applications in signal processing J. Fourier Anal. Appl. 18, 1 (2012) 45-66

### Space-frequency analysis and polynomial spaces

Based on the uncertainty principles developed in my Ph.D. thesis, I started to analyze the properties of space-frequency operators linked to the variances of the uncertainty product. The resulting space-frequency analysis has strong similarities to the space-frequency analysis developed by Landau, Pollak and Slepian. A big advantage of the developed theory is the fact that the localized basis functions can be represented and computed efficiently with orthogonal polynomials. In a second work, this theory was extended successfully for spherical polynomials on the unit sphere.

Fig. 2: Decomposition of a function on the unit sphere with localized polynomial basis functions.
• Erb, W. An orthogonal polynomial analogue of the Landau-Pollak-Slepian time-frequency analysis J. Approx. Theory 166 (2013) 56-77
• Erb, W. and Mathias, S. An alternative to Slepian functions on the unit sphere - A space-frequency analysis based on localized spherical polynomials Appl. Comput. Harmon. Anal. 38, 2 (2015) 222-241

### Iterative solvers for linear inverse problems

Motivated by a collaboration with medics in Lübeck, I started to investigate mathematical problems related to inverse problems in imaging. In a first work, I studied semiiterative methods for the solution of linear ill-posed problems that are constructed using co-dilated orthogonal polynomials. By an appropriate choice of the dilation parameter, the number of iterative steps in the computations can be reduced. In a collaboration with E.V. Semenova, these solvers were combined with adaptive discretization methods.

• Erb, W. Accelerated Landweber methods based on co-dilated orthogonal polynomials Numer. Alg. 68, 2 (2015) 229-260
• Erb, W. and Semenova, E.V. On adaptive discretization schemes for the solution of ill-posed problems with semiiterative methods Appl. Anal. 94, 10 (2015) 2057-2076

### Magnetic Particle Imaging (MPI) and Lissajous curves

In 2014, I initiated together with several colleagues the DFG-funded interdisciplanary scientific network "Development, analysis and application of mathematical methods for Magnetic Particle Imaging (MathMPI) ". In collaboration with C. Kaethner, M. Ahlborg and T.M. Buzug from the Institute of Medical Engineering at the Universität zu Lübeck, I studied methods for the interpolation and processing of data on Lissajous curves. These curves are of particular interest in Magnetic Particle Imaging, since they usually describe the sampling trajectory of the scanning device. In first works, we developed, similar as in the theory of the Padua points, algorithms for an efficient polynomial interpolation on the node points of two-dimensional Lissajous curves. In a joint work with P. Dencker, we were able to extend this theory to multivariate Lissajous-Chebyshev nodes.

Fig. 3: Two Lissajous figures with the respective interpolation node sets.
• Erb, W., Kaethner, C., Ahlborg, M. and Buzug, T.M. Bivariate Lagrange interpolation at the node points of non-degenerate Lissajous curves Numer. Math (2015), accepted for publication, DOI: 10.1007/s00211-015-0762-1
• Erb, W. Bivariate Lagrange interpolation at the node points of Lissajous curves - the degenerate case arXiv:1503.00895v1 [math.NA] (2015)
• Erb, W., Kaethner, C. Dencker, P. and Ahlborg, M. A survey on bivariate Lagrange interpolation on Lissajous nodes Dolomites Research Notes on Approximation 8 (Special issue) (2015), 23-36
• Dencker, P. and Erb, W. Multivariate polynomial interpolation on Lissajous-Chebyshev nodes arXiv:1511.04564v1 [math.NA] (2015)

#### Further links

• CAA: Padova-Verona research group on "Constructive Approximation and Applications"
for further information and references on Padua points and polynomial approximation on Lissajous curves
• Mathematical methods for Magnetic Particle Imaging (MathMPI)
for further information on the DFG-funded interdisciplanary scientific network MathMPI