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**1 - 3**of**3**### Compressed Sensing.

, 2011

"... The central problem of Compressed Sensing is to recover a sparse signal from fewer measurements than its ambient dimension. Recent results by Donoho, and Candes and Tao giving theoretical guarantees that ( 1-minimization succeeds in recovering the signal in a large number of cases have stirred up mu ..."

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The central problem of Compressed Sensing is to recover a sparse signal from fewer measurements than its ambient dimension. Recent results by Donoho, and Candes and Tao giving theoretical guarantees that ( 1-minimization succeeds in recovering the signal in a large number of cases have stirred up much interest in this topic. Subse-quent results followed, where prior information was imposed on the sparse signal and algorithms were proposed and analyzed to incorporate this prior information. In[13] Xu suggested the use of weighted fl-minimization in the case where the additional prior information is probabilistic in nature for a relatively simple probabilistic model. In this thesis, we exploit the techniques developed in [13] to extend the analysis to a more general class of probabilistic models, where the probabilities are evaluations of a continuous function at uniformly spaced points in a given interval. For this case, we use weights which have a similar characterization. We demonstrate our techniques through numerical computations for a certain class of weights and compare some of our results with empirical data obtained through simulations.

### Interpolation via weighted `1 minimization

, 2014

"... Functions of interest are often smooth and sparse in some sense, and both pri-ors should be taken into account when interpolating sampled data. Classical linear interpolation methods are effective under strong regularity assumptions, but cannot incorporate nonlinear sparsity structure. At the same t ..."

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Functions of interest are often smooth and sparse in some sense, and both pri-ors should be taken into account when interpolating sampled data. Classical linear interpolation methods are effective under strong regularity assumptions, but cannot incorporate nonlinear sparsity structure. At the same time, nonlinear methods such as `1 minimization can reconstruct sparse functions from very few samples, but do not necessarily encourage smoothness. Here we show that weighted `1 minimization effectively merges the two approaches, promoting both sparsity and smoothness in reconstruction. More precisely, we provide specific choices of weights in the `1 ob-jective to achieve rates for functions with coefficient sequences in weighted `p spaces, p ≤ 1. We consider the implications of these results for spherical harmonic and poly-nomial interpolation, in the univariate and multivariate setting. Along the way, we extend concepts from compressive sensing such as the restricted isometry property and null space property to accommodate weighted sparse expansions; these developments should be of independent interest in the study of structured sparse approximations and continuous-time compressive sensing problems. Key words: bounded orthonormal systems, compressive sensing, interpolation, weighted sparsity, `1 minimization Dedicated to the memory of Ted Odell 1