Polynomials

Linear functionals

If one considers the set of relevant polynomials as being a vector space or module, it can be useful to consider the “dual space/module” of functions that takes each polynomial to a value in the coefficient ring in a linear way. These “linear functionals” behave like: \(\lambda(\alpha p + \beta q)=\alpha \lambda(p) +\beta \lambda(q)\).

The linear functionals can be combined in a linear way: \((\alpha \lambda + \beta \mu)(p)=\alpha \lambda(p)+\beta\mu(p)\). Sometimes it is helpful to use a “Dirac” notation: \(\lambda(p)=\left< \lambda | p \right>\). This allows one to consider the polynomials to be a set of linear functionals on the dual space/module.

A useful set of functionals projects the polynomials onto their n-th coefficient: \(\left< \pi_n | \sum a_m x^m \right> = a_n \). Other common functionals consist of integrals over intervals with various weight functions.