  Restricted non-linear approximation is a type of N-term approximation where a
measure $\nu$ on the index set (rather than the counting measure) is used to
control the number of terms in the approximation. We show that embeddings for
restricted non-linear approximation spaces in terms of weighted Lorentz
sequence spaces are equivalent to Jackson and Bernstein type inequalities, and
also to the upper and lower Temlyakov property. As applications we obtain
results for wavelet bases in Triebel-Lizorkin spaces by showing the Temlyakow
property in this setting. Moreover, new interpolation results for
Triebel-Lizorkin and Besov spaces are obtained.
