1. Integral-Equation

Hello,

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

Regards

2. Re: Integral-Equation

Originally Posted by Sogan
Hello,

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

Regards
What do you know about $\displaystyle P$, $\displaystyle q$ and $\displaystyle M$ ?

3. Re: Integral-Equation

First P is a polynomial in $\displaystyle \mathbb{R}^n$ 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:

$\displaystyle \int_\mathbb{R}^n \left|P(x)\right|^{-q}*\left|x\right|^{-M} <\infty$

4. 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.

5. 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?

6. Re: Integral-Equation

Originally Posted by Sogan
First P is a polynomial in $\displaystyle \mathbb{R}^n$ 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:

$\displaystyle \int_{\mathbb{R}^n} \left|P(x)\right|^{-q}*\left|x\right|^{-M} <\infty .$
Thanks to this you can apply Fubini theorem.

7. Re: Integral-Equation

But it is a infinite sum? can i use fubini nevertheless?

8. Re: Integral-Equation

Yes, you apply it to the Lebesgue measure on $\displaystyle \mathbb{R}^n$ and the counting measure on $\displaystyle \mathbb{Z}^n$.

9. 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