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I have the following problem at hand. Define the kernel $$K(x_1,x_2) = \int_{-1}^1\int_{-1}^1 \exp(-2\pi\jmath x_1 y_1)R(y_1,y_2)\exp(2\pi\jmath x_2 y_2)\mathrm{d}y_1\mathrm{d_2}.$$

Now, if $R(y_1,y_2)=\delta(y_1-y_2)$ then the resulting Kernel $K(x_1,x_2)$ is a sinc kernel, and as such, has a few dominant eigenvectors (the prolate spherodial wave functions).

My inner kernel $R(y_1,y_2)$ is not a delta-function, but is quite nice (falls of as $(1-|y_1|)(1-|y_2)$). In addition, for $y_1=y_2$ there is a "delta term $1-|y_1|$.

I am not able to solve the integral in closed form. My question is rather loose: what properties of $R(y_1,y_2)$ would guarantee a low dimensionality of $K(y_1,y_2)$. In other words, what properties of the spectrum of $K(y_1,y_2)$ can be deduced from the properties of $R(y_1,y_2)$?

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    $\begingroup$ A possible plan to attack. 1. Divide domain of integration on $x<y$&$x>y$. 2. Take $\frac{1}{x-y}$ as series in these domains. 3. use $\sin(x-y)$. 4. Integrate two one-dimensional integrals in series. At least you will have an answer as some series. $\endgroup$
    – Sergei
    Commented May 17, 2015 at 10:07
  • $\begingroup$ what do you mean by "use sin(x-y)" ?? $\endgroup$
    – Tommy Ding
    Commented May 18, 2015 at 6:25
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    $\begingroup$ $\sin(x-y)=\sin x \cos y - \sin y \cos x$ after that and series expanding for $\frac{1}{x-y}$ the double integral is dividing into two separate in x and in y. $\endgroup$
    – Sergei
    Commented May 18, 2015 at 12:00
  • $\begingroup$ the above comments do not apply after the question was changed $\endgroup$
    – Tommy Ding
    Commented May 19, 2015 at 13:52

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