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Trig polynomials countable?

  1. Sep 13, 2013 #1
    1. The problem statement, all variables and given/known data
    Prove that the set of all trigonometric polynomials with integer coefficients is countable.
    2. Relevant equations
    [itex] t(x)= a+\sum a_ncos(nx)+ \sum b_n sin(nx) [/itex]
    the sum is over n and is from 1 to some natural number.
    3. The attempt at a solution
    So basically we have to look at all the possible trig polynomials of all finite lengths.
    with some natural a out front.
    lets first look at the ones where n=1
    so we have a+cos(1x)+bsin(x)
    How about we map these to the first prime number 2.
    since a can be anything, if a is 1 then this t(x) will go to 2 and if a is 2 then t(x) will go to 2^2
    if a is three then t(x) goes to 2^3.
    Now for the sum from n=1 to 2 will map these to the next prime and do the same process as above with the constant out front. Is this the right idea.
    I am mapping these to prime numbers so we can get a unique mapping and we don't have to worry about sending 2 things to one thing.
     
  2. jcsd
  3. Sep 14, 2013 #2

    Dick

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    In general the set of all finite subsets of a countable set is countable. That's the proof you want. There's nothing about trig polynomials that makes it any easier. Try to prove that first.
     
  4. Sep 14, 2013 #3

    Zondrina

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    Going from the hint Dick has given, start by showing that the set of subsets with no more than ##n## elements is countable. Since you're working finite here, induction should be good I believe.
     
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