  Approximation of high-dimensional functions is a problem in many scientific
fields that is only feasible if advantageous structural properties, such as
sparsity in a given basis, can be exploited. A relevant tool for analysing
sparse approximations is Stechkin's lemma. In its standard form, however, this
lemma does not allow to explain convergence rates for a wide range of relevant
function classes.
  This work presents a new weighted version of Stechkin's lemma that improves
the best $n$-term rates for weighted $\ell^p$-spaces and associated function
classes such as Sobolev or Besov spaces. For the class of holomorphic
functions, which occur as solutions of common high-dimensional
parameter-dependent PDEs, we recover exponential rates that are not directly
obtainable with Stechkin's lemma.
  Since weighted $\ell^p$-summability induces weighted sparsity, compressed
sensing algorithms can be used to approximate the associated functions. To
break the curse of dimensionality, which these algorithms suffer, we recall
that sparse approximations can be encoded efficiently using tensor networks
with sparse component tensors. We also demonstrate that weighted
$\ell^p$-summability induces low ranks, which motivates a second tensor train
format with low ranks and a single weighted sparse core. We present new
alternating algorithms for best $n$-term approximation in both formats.
  To analyse the sample complexity for the new model classes, we derive a novel
result of independent interest that allows the transfer of the restricted
isometry property from one set to another sufficiently close set. Although they
lead up to the analysis of our final model class, our contributions on weighted
Stechkin and the restricted isometry property are of independent interest and
can be read independently.
