  Infinite-dimensional compressed sensing deals with the recovery of analog
signals (functions) from linear measurements, often in the form of integral
transforms such as the Fourier transform. This framework is well-suited to many
real-world inverse problems, which are typically modelled in
infinite-dimensional spaces, and where the application of finite-dimensional
approaches can lead to noticeable artefacts. Another typical feature of such
problems is that the signals are not only sparse in some dictionary, but
possess a so-called local sparsity in levels structure. Consequently, the
sampling scheme should be designed so as to exploit this additional structure.
In this paper, we introduce a series of uniform recovery guarantees for
infinite-dimensional compressed sensing based on sparsity in levels and
so-called multilevel random subsampling. By using a weighted
$\ell^1$-regularizer we derive measurement conditions that are sharp up to log
factors, in the sense they agree with those of certain oracle estimators. These
guarantees also apply in finite dimensions, and improve existing results for
unweighted $\ell^1$-regularization. To illustrate our results, we consider the
problem of binary sampling with the Walsh transform using orthogonal wavelets.
Binary sampling is an important mechanism for certain imaging modalities.
Through carefully estimating the local coherence between the Walsh and wavelet
bases, we derive the first known recovery guarantees for this problem.
