From: lpa@netcom11.netcom.com (Pierre Asselin)
Subject: Re: Product of two generalized polynomials
Date: 3 Feb 2000 09:02:27 GMT
Newsgroups: sci.math.num-analysis
Summary: [missing]

Carlos Freire da Silva Pinto Coelho <cfspc@algos.inesc.pt> writes:

>    Having a polynomial base defined by a recurrence relation P_{k+1}(s)
>= s P_{k}(s)*alfa_k-P_{k-1}(s)*beta_k is there an efficient
>way to determine the decomposition of the product of two polynomials
>P_g*P_h as a sum of polynomial in the same base?

Your recurrence relation means that, in the basis [P_0(s), P_1(s), ...]
the operator "multiplication by s" has the matrix

        [   0       -beta_1/alfa_1         0               0        ... ]
        [ 1/alfa_0         0        -beta_2/alfa_2         0        ... ]
 [s] =  [   0          1/alfa_1            0        -beta_3/alfa_3  ... ]
        [   0              0            1/alfa_2           0        ... ]
        [  ...            ...             ...             ...       ... ]

You can then use the recurrence relation again to find the matrix of
the operator "multiplication by P_k":

    [P_k+1] = alfa_{k} [P_{k}]*[s] + beta_{k} [P_{k-1}]  .

All the matrices commute, since the multiplication operators commute.
--
--Pierre Asselin, Westminster, Colorado
  lpa@netcom.com