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Non-linear convolution and power series

  1. Aug 27, 2013 #1
    1. The problem statement, all variables and given/known data

    Hi, suppose we have the summation
    [tex] \sum_{i=0}^{n-1} \sum_{j=0}^{n-1} a_j b_{i-j}^{j} x^i,[/tex]

    where the subscripts are taken modulo [itex]n[/itex], and [itex]a_i^n = a_i, b_i^n = b_i[/itex] for each [itex]i[/itex].

    Write the above power series as a product of two power series modulo [itex]x^n - x[/itex].


    2. Relevant equations

    I'm only aware of the regular Cauchy (linear) convolution. That is,
    [tex] \sum_{i=0}^{n-1}\sum_{j=0}^{n-1}a_j b_{i-j}x^i = \left( \sum_{i=0}^{n-1}a_i x^i \right) \left( \sum_{j=0}^{n-1}b_j x^j \right).[/tex]

    3. The attempt at a solution
    I'm frankly not sure...

    Thanks!
     
    Last edited: Aug 27, 2013
  2. jcsd
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