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Thread: using periodicity of a funtion

  1. #1
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    using periodicity of a funtion

    Hello!
    I got the following function:

    $\displaystyle f_{i} (x) = \int_{0}^{x} (-1)^{\lfloor{t \cdot 2^{i}}\rfloor} \ dt$

    The function has a periodicity of $\displaystyle 2^{1-i}$.

    Let $\displaystyle x_{0}=0.b_{1}b_{2}b_{3} \cdots$ be the binary expansion of a number between 0 and 1.

    Letīs consider $\displaystyle i > n$.

    If we look at
    $\displaystyle f_{i} (0.b_{1}b_{2}b_{3} \cdots)$ why do we have by the periodicity of $\displaystyle f_{i}$ for i > n
    $\displaystyle f_{i} (0.b_{1}b_{2}b_{3} \cdots) = f_{i} (0.000....0b_{n+1}b_{n+2}b_{n+3} \cdots) $ ?

    I guess that the period is subtracted exactly i times, but I can not show it.
    Can anyone help please? Thanks.
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  2. #2
    Super Member girdav's Avatar
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    Re: using periodicity of a funtion

    For $\displaystyle n<i$, we have $\displaystyle f\left(\sum_{k=1}^{+\infty}b_k2^{-k}\right) = f\left(\sum_{k=1}^nb_k2^{-k} +\sum_{k=n+1}^{+\infty}b_k2^{-k}\right)$ and you have to show that $\displaystyle \sum_{k=1}^nb_k2^{-k} =j2^{1-i}$ for an integer $\displaystyle j$.
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  3. #3
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    Re: using periodicity of a funtion

    Quote Originally Posted by girdav View Post
    For $\displaystyle n<i$, we have $\displaystyle f\left(\sum_{k=1}^{+\infty}b_k2^{-k}\right) = f\left(\sum_{k=1}^nb_k2^{-k} +\sum_{k=n+1}^{+\infty}b_k2^{-k}\right)$ and you have to show that $\displaystyle \sum_{k=1}^nb_k2^{-k} =j2^{1-i}$ for an integer $\displaystyle j$.
    Your advice has been great!
    What is left to show is not equivalent to show that
    $\displaystyle 2^{i-1} \cdot \sum_{k=1}^{n} b_{k} 2^{-k}$ is an integer, correct?. I guess I actually have to calculate that integer j?

    Thanks.
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  4. #4
    Super Member girdav's Avatar
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    Re: using periodicity of a funtion

    You have to show that $\displaystyle 2^{i-1} \cdot \sum_{k=1}^{n} b_{k} 2^{-k} = \sum_{k=1}^{n} b_{k} 2^{i-1-k}$ is an integer. Use the fact that $\displaystyle k\leq n<i$
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  5. #5
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    Re: using periodicity of a funtion

    Quote Originally Posted by girdav View Post
    You have to show that $\displaystyle 2^{i-1} \cdot \sum_{k=1}^{n} b_{k} 2^{-k} = \sum_{k=1}^{n} b_{k} 2^{i-1-k}$ is an integer. Use the fact that $\displaystyle k\leq n<i$
    Yeah! That is what I did. I just wasnīt sure I had to show exactly that one cause I first thought I had to calculate a specific j.

    I know that $\displaystyle i-k > 0$ and then $\displaystyle i-k-1 \geq 0$. Thus the sum will be an integer.
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