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How do I prove the following?

$$\int\det\left(K(x_{i},x_{j})\right)_{1\leq i,j\leq n}dx_{1} \cdots dx_{N}=\underset{i=1}{\overset{n}{\prod}}\left(\int K(x_{i},x_{i})\;dx_{i}-(i-1)\right)$$

where

$$K(x,y)=\sum_{l=1}^n \psi_l(x)\overline{\psi_l}(y)$$ and $$\{\psi_l(x)\}_{l=1}^n$$ is an ON-sequence in $L^2$. One may note that $$\int K(x_i,x_j)K(x_j,x_i) \; d\mu(x_i)=K(x_j,x_j)$$ and also that $$\int K(x_a,x_b)K(x_b,x_c)d\mu(x_b)=K(x_a,x_c).$$

  • 0
    Is there a link between $n$ and $N$?2011-08-27
  • 0
    yes, my bad. n=N2011-08-27

3 Answers 3