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Math Help - Non dyadic Values

  1. #1
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    Non dyadic Values

    Let 0 < x < 2 be a non-dyadic value expressed as x =summation(d_n/2^n)
    ; dn = 0 or 1 for all n.
    Show that for gm defined as gm=summation(1/2^nh(2^nx) from 0 to m one has
    g'm(x) = (-1)^d0 + (-1)^d1 + .... + (-1)^dm
    where h=|x|.



    I started looking at some example. First, I had dn=0 if n was odd and 1 if n was even.
    Then x=1/4^n=4/3
    Alternatively dn=0 if n is even and 1 if n is odd.
    Then x=1/2^(2n+1)=2/3
    So g_0'(4/3)=h'(4/3)=1
    g_0'(2/3)=h'(2/3)=1
    g_1'(4/3)=h'(4/3)+h'(2*4/3)=-1+h'(2+2/3)=-1+h'(2/3)=-1+1=0
    g_1'(2/3)=0
    Then g_1'(x)=h'(summ(dn/2^n)+h'(2*sum(dn/2^n))
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  2. #2
    MHF Contributor Drexel28's Avatar
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    Quote Originally Posted by kathrynmath View Post
    Let 0 < x < 2 be a non-dyadic value expressed as x =summation(d_n/2^n)
    ; dn = 0 or 1 for all n.
    Show that for gm defined as gm=summation(1/2^nh(2^nx) from 0 to m one has
    g'm(x) = (-1)^d0 + (-1)^d1 + .... + (-1)^dm
    where h=|x|.



    I started looking at some example. First, I had dn=0 if n was odd and 1 if n was even.
    Then x=1/4^n=4/3
    Alternatively dn=0 if n is even and 1 if n is odd.
    Then x=1/2^(2n+1)=2/3
    So g_0'(4/3)=h'(4/3)=1
    g_0'(2/3)=h'(2/3)=1
    g_1'(4/3)=h'(4/3)+h'(2*4/3)=-1+h'(2+2/3)=-1+h'(2/3)=-1+1=0
    g_1'(2/3)=0
    Then g_1'(x)=h'(summ(dn/2^n)+h'(2*sum(dn/2^n))
    I'm sorry, there is way to little info here to see what's going on. Also, you might want to tex this, it's hard to read.
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