Given polynomials p and q expressed in a Chebyshev basis, form the product $pq$ expressed in a Chebyshev basis.

All Chebyshev basis polynomials $T\left(k\,x\right)$ which appear must have the same second argument x (which can be any expression).

The input polynomials must be in expanded form (i.e. a sum of products). Normally, each term in the sum contains one and only one $T\left(k\,x\right)$ factor except that if there are terms in the sum containing no $T\left(k\,x\right)$ factor then each such term t is interpreted to represent $tT\left(0\,x\right)$ provided that t and x have no variables in common.

If no $T\left(k\,x\right)$ factor appears in p or in q then the ordinary product $pq$ is returned.

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

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

$\mathrm{Digits}≔3\:$

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

$a≔0.880T\left(1\,x\right)-0.0391T\left(3\,x\right)+0.000500T\left(5\,x\right)$

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

$b≔1.26T\left(0\,x\right)+1.13T\left(1\,x\right)+0.271T\left(2\,x\right)+0.0443T\left(3\,x\right)+0.00547T\left(4\,x\right)+0.000543T\left(5\,x\right)$

$\mathrm{chebmult}\left(a\,b\right)$

$0.496T\left(0\,x\right)+1.22T\left(1\,x\right)+0.494T\left(2\,x\right)+0.0718T\left(3\,x\right)-0.00212T\left(4\,x\right)-0.00227T\left(5\,x\right)-0.000344T\left(6\,x\right)-0.0000390T\left(7\,x\right)+5.\times {10}^{\mathrm{-7}}T\left(8\,x\right)+1.37\times {10}^{\mathrm{-6}}T\left(9\,x\right)+1.36\times {10}^{\mathrm{-7}}T\left(10\,x\right)$

$c≔\mathrm{c0}T\left(0\,x\right)+\mathrm{c1}T\left(1\,x\right)$

$c≔\mathrm{c0}T\left(0\,x\right)+\mathrm{c1}T\left(1\,x\right)$

$d≔\mathrm{d0}T\left(0\,x\right)+\mathrm{d1}T\left(1\,x\right)$

$d≔\mathrm{d0}T\left(0\,x\right)+\mathrm{d1}T\left(1\,x\right)$

$\mathrm{chebmult}\left(c\,d\right)$

$\left(\mathrm{c0}\mathrm{d0}+\frac{\mathrm{d1}\mathrm{c1}}{2}\right)T\left(0\,x\right)+\left(\mathrm{c0}\mathrm{d1}+\mathrm{d0}\mathrm{c1}\right)T\left(1\,x\right)+\frac{\mathrm{d1}\mathrm{c1}T\left(2\,x\right)}{2}$

$\mathrm{chebmult}\left(T\left(j\,x\right)\,T\left(k\,x\right)\right)$

$\frac{T\left(\left|-k+j\right|\,x\right)}{2}+\frac{T\left(k+j\,x\right)}{2}$

$\mathrm{assume}\left(0<j\&comma;j<k\right)$

$\mathrm{chebmult}\left(\mathrm{c0}+\mathrm{cj}T\left(j\&comma;x\right)\&comma;T\left(k\&comma;x\right)\right)$

$\frac{\mathrm{cj}T\left(\mathrm{k~}-\mathrm{j~}\&comma;x\right)}{2}+\mathrm{c0}T\left(\mathrm{k~}\&comma;x\right)+\frac{\mathrm{cj}T\left(\mathrm{j~}+\mathrm{k~}\&comma;x\right)}{2}$

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

$e≔a+\mathrm{ck}T\left(k\&comma;x\right)$

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

$\mathrm{chebmult}\left(e\&comma;T\left(j\&comma;x\right)\right)$

$0.500\mathrm{ck}T\left(\mathrm{k~}-\mathrm{j~}\&comma;x\right)+0.000250T\left(\mathrm{j~}-5\&comma;x\right)-0.0196T\left(\mathrm{j~}-3\&comma;x\right)+0.440T\left(\mathrm{j~}-1\&comma;x\right)+0.440T\left(1+\mathrm{j~}\&comma;x\right)-0.0196T\left(3+\mathrm{j~}\&comma;x\right)+0.000250T\left(5+\mathrm{j~}\&comma;x\right)+0.500\mathrm{ck}T\left(\mathrm{k~}+\mathrm{j~}\&comma;x\right)$