  We provide novel sufficient conditions for the uniform recovery of sparse
Legendre expansions using $\ell_1$ minimization, where the sampling points are
drawn according to orthogonalization (uniform) measure. So far, conditions of
the form $m \gtrsim \Theta^2 s \times \textit{log factors}$ have been relied on
to determine the minimum number of samples $m$ that guarantees successful
reconstruction of $s$-sparse vectors when the measurement matrix is associated
to an orthonormal system. However, in case of sparse Legendre expansions, the
uniform bound $\Theta$ of Legendre systems is so high that these conditions are
unable to provide meaningful guarantees. In this paper, we present an analysis
which employs the envelop bound of all Legendre polynomials instead, and prove
a new recovery guarantee for $s$-sparse Legendre expansions, $$ m \gtrsim {s^2}
\times \textit{log factors}, $$ which is independent of $\Theta$. Arguably,
this is the first recovery condition established for orthonormal systems
without assuming the uniform boundedness of the sampling matrix. The key
ingredient of our analysis is an extension of chaining arguments, recently
developed in [Bou14,CDTW15], to handle the envelope bound. Furthermore, our
recovery condition is proved via restricted eigenvalue property, a less
demanding replacement of restricted isometry property which is perfectly suited
to the considered scenario. Along the way, we derive simple criteria to detect
good sample sets. Our numerical tests show that sets of uniformly sampled
points that meet these criteria will perform better recovery on average.
