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Math Help - Integral-Equation

  1. #1
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    Integral-Equation

    Hello,

    I it right to change the summation with the integral sign: _limits_{[0,1]^n}\sum_{k \in \mathbb{Z}^n}(\left|P(k+\alpha)\right|^{-q}*\left|k\right|^{-M}) d\alpha <=\sum_{k \in \mathbb{Z}^n}\int_limits_{[0,1]^n}(\left|P(k+\alpha)\right|^{-q}*\left|k\right|^{-M}) d\alpha " alt="\int_limits_{[0,1]^n}\sum_{k \in \mathbb{Z}^n}(\left|P(k+\alpha)\right|^{-q}*\left|k\right|^{-M}) d\alpha <=\sum_{k \in \mathbb{Z}^n}\int_limits_{[0,1]^n}(\left|P(k+\alpha)\right|^{-q}*\left|k\right|^{-M}) d\alpha " />


    Regards
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  2. #2
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    Re: Integral-Equation

    Quote Originally Posted by Sogan View Post
    Hello,

    I it right to change the summation with the integral sign: \int_{[0,1]^n}\sum_{k \in \mathbb{Z}^n}(\left|P(k+\alpha)\right|^{-q}*\left|k\right|^{-M}) d\alpha \leq \sum_{k \in \mathbb{Z}^n}\int_{[0,1]^n}(\left|P(k+\alpha)\right|^{-q}*\left|k\right|^{-M}) d\alpha


    Regards
    What do you know about P, q and M ?
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  3. #3
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    Re: Integral-Equation

    First P is a polynomial in \mathbb{R}^n of order m. 0<q<1/m and M>0 is some constant.

    Furthermore we know the following fact: M>0 is chosen s.t. the integral is finite:

     <br />
\int_\mathbb{R}^n \left|P(x)\right|^{-q}*\left|x\right|^{-M} <\infty<br />
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  4. #4
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    Re: Integral-Equation

    You can change between the summation and the integral if the integrand is uniformly convergent. Which means the sum has to be uniformly convergent.
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  5. #5
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    Re: Integral-Equation

    Ok. The set, over which we integrate is compact. Therefore it is sufficient, that the sum is just convergent for all points right?
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  6. #6
    Super Member girdav's Avatar
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    Re: Integral-Equation

    Quote Originally Posted by Sogan View Post
    First P is a polynomial in \mathbb{R}^n of order m. 0<q<1/m and M>0 is some constant.

    Furthermore we know the following fact: M>0 is chosen s.t. the integral is finite:

     \int_{\mathbb{R}^n} \left|P(x)\right|^{-q}*\left|x\right|^{-M} <\infty .
    Thanks to this you can apply Fubini theorem.
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  7. #7
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    Re: Integral-Equation

    But it is a infinite sum? can i use fubini nevertheless?
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  8. #8
    Super Member girdav's Avatar
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    Re: Integral-Equation

    Yes, you apply it to the Lebesgue measure on \mathbb{R}^n and the counting measure on \mathbb{Z}^n.
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  9. #9
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    Re: Integral-Equation

    Can you please tell me in more detail, what you really mean? I don't see it. I don't se a counting measure, for instance. You are right, there is a L-measure. And furthermore some series of functions (whereas i don't know really, whether it is convergent or not)

    Thank you very much
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