  The approximation numbers of the $L_2$-embedding of mixed order Sobolev
functions on the $d$-torus are well studied. They are given as the
nonincreasing rearrangement of the $d$-th tensor power of the approximation
number sequence in the univariate case. I present results on the asymptotic and
preasymptotic behavior for tensor powers of arbitrary sequences of polynomial
decay. This can be used to study the approximation numbers of many other tensor
product operators, like the embedding of mixed order Sobolev functions on the
$d$-cube into $L_2\left([0,1]^d\right)$ or the embedding of mixed order Jacobi
functions on the $d$-cube into $L_2\left([0,1]^d,w_d\right)$ with Jacobi weight
$w_d$.
