Page 107 - Textos de Matemática Vol. 34

ORTHOGONAL POLYNOMIALS 97
This motivates the next definitions (which agree with the usual ones in the sense of the Theory of Distributions).
Definition4.1. Letu∈P′,g∈P,andc∈C. Wedefine
(i) the left-multiplication of u by g: it is the functional in P′, denoted by gu,
such that
⟨gu,f⟩:=⟨u,gf⟩, f∈P;
(ii) the derivative of u: it is the functional in P′, denoted by Du, such that
⟨Du,f⟩ := −⟨u,Df⟩ , f ∈ P ;
(iii) the division of u by x − c: it is the functional in P′, denoted by (x − c)−1u,
such that
⟨(x−c)−1u,f⟩:=⟨u,θcf⟩, f∈P.
Notice that the derivative of the left-multiplication of a functional by a polyno-
mial fulfils the usual rule
D(gu)=g′u+gDu, g∈P,u∈P′.
We have defined the left-multiplication of a linear functional by a polynomial in such a way that it is another linear functional. We can also define a right- multiplication of a linear functional by a polynomial, but in this case we get a polynomial.
Definition 4.2. Let u ∈ P′ and f ∈ P. The right-multiplication of u by f is the polynomial, denoted by uf, defined by
uf(x) := ⟨uξ, θξ(xf)⟩ ,
where the subscript ξ in uξ means that u acts on polynomials in the variable ξ.
Putting f(x) = nν=0 aνxν , this polynomial uf can be explicitly given by
nn (4.1) uf(x)= aνui−ν xν ,
ν=0 i=ν
and it also admits the very useful matrix representation
u0 0 0···01
u1u0 0···0x
u u u ···0x2 uf(x)=[a0a1 ···an] 2 1 0  .
 . . . ... .  .  
un un−1 un−2 ··· u0 xn