The input expression e is assumed to be a polynomial expressed in terms of a Chebyshev basis $T\left(0\,x\right),...$.

First the expression e is collected in 'T'. Then the terms in the collected polynomial expression are sorted in ``Chebyshev order'';  i.e. the $T\left(k\,x\right)$ basis polynomials are ordered in ascending order with respect to the first argument.

If some basis polynomials $T\left(k\,x\right)$ have non-numeric first argument then ordering will be attempted using the ``is'' predicate. If that is not successful then ordering is performed only with respect to numeric first arguments (other terms are left as trailing terms).

Note that chebsort is a destructive operation because it invokes the Maple sort function (see sort);  i.e. the input expression is sorted ``in-place''.

The command with(numapprox,chebsort) allows the use of the abbreviated form of this command.

$\mathrm{with}\left(\mathrm{numapprox}\right)\:$

$\mathrm{Digits}≔3\:$

$a≔\mathrm{chebyshev}\left(\sin \left(x\right)\,x\right)\:$

$b≔\mathrm{chebyshev}\left(\cos \left(x\right)\,x\right)\:$

$c≔a+b$

$c≔0.880T\left(1\,x\right)-0.0391T\left(3\,x\right)+0.000500T\left(5\,x\right)+0.765T\left(0\,x\right)-0.230T\left(2\,x\right)+0.00495T\left(4\,x\right)$

$\mathrm{chebsort}\left(c\right)$

$0.765T\left(0\,x\right)+0.880T\left(1\,x\right)-0.230T\left(2\,x\right)-0.0391T\left(3\,x\right)+0.00495T\left(4\,x\right)+0.000500T\left(5\,x\right)$

$\mathrm{assume}\left(5<j\&comma;j<k\right)$

$d≔1.2y+\mathrm{cj}T\left(j\&comma;x\right)+a+\mathrm{ck}T\left(k\&comma;x\right)$

$d≔1.2y+\mathrm{cj}T\left(\mathrm{j~}\&comma;x\right)+0.880T\left(1\&comma;x\right)-0.0391T\left(3\&comma;x\right)+0.000500T\left(5\&comma;x\right)+\mathrm{ck}T\left(\mathrm{k~}\&comma;x\right)$

$\mathrm{chebsort}\left(d\right)$

$0.880T\left(1\&comma;x\right)-0.0391T\left(3\&comma;x\right)+0.000500T\left(5\&comma;x\right)+\mathrm{cj}T\left(\mathrm{j~}\&comma;x\right)+\mathrm{ck}T\left(\mathrm{k~}\&comma;x\right)+1.2y$