arXiv subjects

Holger Rauhut

Publications and source records attributed to Holger Rauhut.

78 records · Page 5Linked to original sources

arXiv2005

Random Sampling of Sparse Trigonometric Polynomials

We study the problem of reconstructing a multivariate trigonometric polynomial having only few non-zero coefficients from few random samples. Inspired by recent work of Candes, Romberg and Tao we propose to recover the polynomial by Basis Pursuit, i.e., by $\ell^1$-minimization. Numerical experiments show that in many cases the trigonometric polynomial can be recovered exactly provided the number $N$ of samples is high enough compared to the ``sparsity'' -- the number of non-vanishing coefficients. However, $N$ can be chosen small compared to the assumed maximal degree of the trigonometric polynomial. Hence, the proposed scheme may overcome the Nyquist rate. We present two theorems that explain this observation. Unexpectly, they establish a connection to an interesting combinatorial problem concerning set partitions, which seemingly has not yet been considered before.

math.CA↗

arXiv2005

Wiener Amalgam Spaces with respect to Quasi-Banach Spaces

We generalize the theory of Wiener amalgam spaces on locally compact groups to quasi-Banach spaces. As a main result we provide convolution relations for such spaces. Also we weaken the technical assumption that the global component is invariant under right translations, which is even new for the classical Banach space case. To illustrate our theory we discuss in detail an example on the $ax+b$ group.

math.FA↗

arXiv2005

Coorbit Space Theory for Quasi-Banach Spaces

We generalize the classical coorbit space theory developed by Feichtinger and Gr"ochenig to quasi-Banach spaces. As a main result we provide atomic decompositions for coorbit spaces defined with respect to quasi-Banach spaces. These atomic decompositions are used to prove fast convergence rates of best $n$-term approximation schemes. We apply the abstract theory to time-frequency analysis of modulation spaces $M^{p,q}_m$, $0< p,q \leq \infty$.

math.FA↗

arXiv2005

Radial Time-Frequency Analysis and Embeddings of Radial Modulation Spaces

In this paper we construct frames of Gabor type for the space $L^2_{rad}(\R^d)$ of radial $L^2$-functions, and more generally, for subspaces of modulation spaces consisting of radial distributions. Hereby, each frame element itself is a radial function. This construction is based on a generalization of the so called Feichtinger-Gröchenig theory -- sometimes also called coorbit space theory -- which was developed in an earlier article. We show that this new type of Gabor frames behaves better in linear and non-linear approximation in a certain sense than usual Gabor frames when approximating a radial function. Moreover, we derive new embedding theorems for coorbit spaces restricted to invariant vectors (functions) and apply them to modulation spaces of radial distributions. As a special case this result implies that the Feichtinger algebra $(S_0)_{rad}(\R^d) = M^1_{rad}(\R^d)$ restricted to radial functions is embedded into the Sobolev space $H^{(d-1)/2}_{rad}(\R^d)$. Moreover, for $d\geq 2$ the embedding $(S_0)_{rad}(\R^d) \hookrightarrow L^2_{rad}(\R^d)$ is compact.

math.FA↗

arXiv2004

Continuous Frames, Function Spaces, and the Discretization Problem

A continuous frame is a family of vectors in a Hilbert space which allows reproductions of arbitrary elements by continuous superpositions. Associated to a given continuous frame we construct certain Banach spaces. Many classical function spaces can be identified as such spaces. We provide a general method to derive Banach frames and atomic decompositions for these Banach spaces by sampling the continuous frame. This is done by generalizing the coorbit space theory developed by Feichtinger and Groechenig. As an important tool the concept of localization of frames is extended to continuous frames. As a byproduct we give a partial answer to the question raised by Ali, Antoine and Gazeau whether any continuous frame admits a corresponding discrete realization generated by sampling.

math.FA↗

arXiv2003

Radial multiresolution in dimension three

We present a construction of a wavelet-type orthonormal basis for the space of radial $L^2$-functions in $\R^3$ via the concept of a radial multiresolution analysis. The elements of the basis are obtained from a single radial wavelet by usual dilations and generalized translations. Hereby the generalized translation reveals the group convolution of radial functions in $\R^3$. We provide a simple way to construct a radial scaling function and a radial wavelet from an even classical scaling function on $\R$. Furthermore, decomposition and reconstruction algorithms are formulated.

math.FA↗