Trigonometric polynomials are widely used for the approximation of a smooth function $f$ from a set of nonuniformly spaced samples $\{f(x_j)\}_{j=0}^{N-1}$. If the samples are perturbed by noise, controlling the smoothness of the trigonometric approximation becomes an essential issue to avoid overfitting and underfitting of the data. Using the polynomial degree as regularization parameter we derive a multi-level algorithm that iteratively adapts to the least squares solution of optimal smoothness. The proposed algorithm computes the solution in at most $\cal{O}(NM + M^2)$ operations ($M$ being the polynomial degree of the approximation) by solving a family of nested Toeplitz systems. It is shown how the presented method can be extended to multivariate trigonometric approximation. We demonstrate the performance of the algorithm by applying it in echocardiography to the recovery of the boundary of the Left Ventricle.

We give an overview of recent developments in the problem of reconstructing a band-limited signal from non-uniform sampling from a numerical analysis view point. It is shown that the appropriate design of the finite-dimensional model plays a key role in the numerical solution of the non-uniform sampling problem. In the one approach (often proposed in the literature) the finite-dimensional model leads to an ill-posed problem even in very simple situations. The other approach that we consider leads to a well-posed problem that preserves important structural properties of the original infinite-dimensional problem and gives rise to efficient numerical algorithms. Furthermore a fast multilevel algorithm is presented that can reconstruct signals of unknown bandwidth from noisy non-uniformly spaced samples. We also discuss the design of efficient regularization methods for ill-conditioned reconstruction problems. Numerical examples from spectroscopy and exploration geophysics demonstrate the performance of the proposed methods.

A shift-invariant system is a collection of functions $\{g_{m,n}\}$ of the form $g_{m,n}(k) = g_m(k-an)$. Such systems play an important role in time-frequency analysis and digital signal processing. A principal problem is to find a dual system $\gamma_{m,n}(k) = \gamma_m(k-an)$ such that each function $f$ can be written as $f = \sum < f, \gamma_{m,n} > g_{m,n}$. The mathematical theory usually addresses this problem in infinite dimensions (typically in $L_2(R)$ or $l_2(Z)$), whereas numerical methods have to operate with a finite-dimensional model. Exploiting the link between the frame operator and Laurent operators with matrix-valued symbol, we apply the finite section method to show that the dual functions obtained by solving a finite-dimensional problem converge to the dual functions of the original infinite-dimensional problem in $l_2(Z)$. For compactly supported $g_{m,n}$ (FIR filter banks) we prove an exponential rate of convergence and derive explicit expressions for the involved constants. Further we investigate under which conditions one can replace the discrete model of the finite section method by the periodic discrete model, which is used in many numerical procedures. Again we provide explicit estimates for the speed of convergence. Some remarks on tight frames complete the paper.

We consider three problems for Gabor frames that have recently received much attention. The first problem concerns the approximation of dual Gabor frames in $L_2(R)$ by finite-dimensional methods. Utilizing Wexler-Raz type duality relations we derive a method to approximate the dual Gabor frame, that is much simpler than previously proposed techniques. Furthermore it enables us to give estimates for the approximation rate when the dimension of the finite model approaches infinity. The second problem concerns the relation between the decay of the window function $g$ and its dual $\gamma$. Based on results on commutative Banach algebras and Laurent operators we derive a general condition under which the dual $\gamma$ inherits the decay properties of $g$. The third problem concerns the design of pulse shapes for orthogonal frequency division multiplex (OFDM) systems for time- and frequency dispersive channels. In particular, we provide a theoretical foundation for a recently proposed algorithm to construct orthogonal transmission functions that are well localized in the time-frequency plane.

The stories told in this paper are dealing with the solution of finite, infinite, and biinfinite Toeplitz-type systems. A crucial role plays the off-diagonal decay behavior of Toeplitz matrices and their inverses. Classical results of Gelfand et al. on commutative Banach algebras yield a general characterization of this decay behavior. We then derive estimates for the approximate solution of (bi)infinite Toeplitz systems by the finite section method, showing that the approximation rate depends only on the decay of the entries of the Toeplitz matrix and its condition number. Furthermore, we give error estimates for the solution of doubly infinite convolution systems by finite circulant systems. Finally, some quantitative results on the construction of preconditioners via circulant embedding are derived, which allow to provide a theoretical explanation for numerical observations made by some researchers in connection with deconvolution problems.

Is compressive sensing overrated? Or can it live up to our expectations? What will come after compressive sensing and sparsity? And what has Galileo Galilei got to do with it? Compressive sensing has taken the signal processing community by storm. A large corpus of research devoted to the theory and numerics of compressive sensing has been published in the last few years. Moreover, compressive sensing has inspired and initiated intriguing new research directions, such as matrix completion. Potential new applications emerge at a dazzling rate. Yet some important theoretical questions remain open, and seemingly obvious applications keep escaping the grip of compressive sensing. In this paper I discuss some of the recent progress in compressive sensing and point out key challenges and opportunities as the area of compressive sensing and sparse representations keeps evolving. I also attempt to assess the long-term impact of compressive sensing.

For a given class ${\cal F}$ of uniform frames of fixed redundancy we define a Grassmannian frame as one that minimizes the maximal correlation $|< f_k,f_l >|$ among all frames $\{f_k\}_{k \in {\cal I}} \in {\cal F}$. We first analyze finite-dimensional Grassmannian frames. Using links to packings in Grassmannian spaces and antipodal spherical codes we derive bounds on the minimal achievable correlation for Grassmannian frames. These bounds yield a simple condition under which Grassmannian frames coincide with uniform tight frames. We exploit connections to graph theory, equiangular line sets, and coding theory in order to derive explicit constructions of Grassmannian frames. Our findings extend recent results on uniform tight frames. We then introduce infinite-dimensional Grassmannian frames and analyze their connection to uniform tight frames for frames which are generated by group-like unitary systems. We derive an example of a Grassmannian Gabor frame by using connections to sphere packing theory. Finally we discuss the application of Grassmannian frames to wireless communication and to multiple description coding.

We investigate pseudodifferential operators on arbitrary locally compact abelian groups. As symbol classes for the Kohn-Nirenberg calculus we introduce a version of Sjoestrand's class. Pseudodifferential operators with such symbols form a Banach algebra that is closed under inversion. Since "hard analysis" techniques are not available on locally compact abelian groups, a new time-frequency approach is used with the emphasis on modulation spaces, Gabor frames, and Banach algebras of matrices. Sjoestrand's original results are thus understood as a phenomenon of abstract harmonic analysis rather than "hard analysis" and are proved in their natural context and generality.

For single-carrier systems with frequency domain equalization, decision feedback equalization (DFE) performs better than linear equalization and has much lower computational complexity than sequence maximum likelihood detection. The main challenge in DFE is the feedback symbol selection rule. In this paper, we give a theoretical framework for a simple, sparsity based thresholding algorithm. We feed back multiple symbols in each iteration, so the algorithm converges fast and has a low computational cost. We show how the initial solution can be obtained via convex relaxation instead of linear equalization, and illustrate the impact that the choice of the initial solution has on the bit error rate performance of our algorithm. The algorithm is applicable in several existing wireless communication systems (SC-FDMA, MC-CDMA, MIMO-OFDM). Numerical results illustrate significant performance improvement in terms of bit error rate compared to the MMSE solution.

Let $\mathbb{H}$ be the general, reduced Heisenberg group. Our main result establishes the inverse-closedness of a class of integral operators acting on $L^{p}(\mathbb{H})$, given by the off-diagonal decay of the kernel. As a consequence of this result, we show that if $\alpha_{1}I+S_{f}$, where $S_{f}$ is the operator given by convolution with $f$, $f\in L^{1}_{v}(\mathbb{H})$, is invertible in $\B(L^{p}(\mathbb{H}))$, then (\alpha_{1}I+S_{f})^{-1}=\alpha_{2}I+S_{g}$, and $g\in L^{1}_{v}(\mathbb{H})$. We prove analogous results for twisted convolution operators and apply the latter results to a class of Weyl pseudodifferential operators. We briefly discuss relevance to mobile communications.