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Tommy Ding
Tommy Ding
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integral over sinc-kernel Transformation of kernel

I have the following integral that I need to solve, but cannot make much progress. Anyone that have seen it before?

$$K(A,B) = \int_{-1}^1\int_{-1}^1 \cos(2\pi Ax)\cos(2\pi By)\mathrm{sinc}((x-y))(1-|x|)(1-|y|)\mathrm{d}x\mathrm{d}y.$$

The parameters A and B are less than 0.5 in magnitudeproblem at hand. Define the kernel $$K(x_1,x_2) = \int_{-1}^1\int_{-1}^1 \exp(-2\pi\jmath x_1 y_1)R(y_1,y_2)\exp(2\pi\jmath x_2 y_2)\mathrm{d}y_1\mathrm{d_2}.$$

In factNow, the end result I would like to obtain is that the number of significant dimensions in the kernelif $K(A,B)$ is small$R(y_1,y_2)=\delta(y_1-y_2)$ then the resulting Kernel (This$K(x_1,x_2)$ is true btw). For thea sinc kernel, the eigenfunctions areand as such, has a few dominant eigenvectors (the prolate spheroidalspherodial wave functions, and it).

My inner kernel $R(y_1,y_2)$ is known fromnot a delta-function, e.gbut is quite nice (falls of as $(1-|y_1|)(1-|y_2)$). In addition, Slepian's work that onlyfor $y_1=y_2$ there is a few of them are sufficent"delta term $1-|y_1|$.

I am not able to representsolve the Kernel wellintegral in closed form. My question is rather loose: Now, for the modified Kernelwhat properties of $K(A,B)$ I$R(y_1,y_2)$ would like to obtainguarantee a similar resultlow 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 $R(y_1,y_2)$?

integral over sinc-kernel

I have the following integral that I need to solve, but cannot make much progress. Anyone that have seen it before?

$$K(A,B) = \int_{-1}^1\int_{-1}^1 \cos(2\pi Ax)\cos(2\pi By)\mathrm{sinc}((x-y))(1-|x|)(1-|y|)\mathrm{d}x\mathrm{d}y.$$

The parameters A and B are less than 0.5 in magnitude.

In fact, the end result I would like to obtain is that the number of significant dimensions in the kernel $K(A,B)$ is small (This is true btw). For the sinc kernel, the eigenfunctions are prolate spheroidal wave functions, and it is known from, e.g., Slepian's work that only a few of them are sufficent to represent the Kernel well. Now, for the modified Kernel $K(A,B)$ I would like to obtain a similar result.

Transformation of kernel

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 $R(y_1,y_2)$?

typo in formula
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Tommy Ding
Tommy Ding
  • 21
  • 2

I have the following integral that I need to solve, but cannot make much progress. Anyone that have seen it before?

$$K(A,B) = \int_{-1}^1\int_{-1}^1 \cos(2\pi Ax)\cos(2\pi Bx)\mathrm{sinc}((x-y))(1-|x|)(1-|y|)\mathrm{d}x\mathrm{d}y.$$$$K(A,B) = \int_{-1}^1\int_{-1}^1 \cos(2\pi Ax)\cos(2\pi By)\mathrm{sinc}((x-y))(1-|x|)(1-|y|)\mathrm{d}x\mathrm{d}y.$$

The parameters A and B are less than 0.5 in magnitude.

In fact, the end result I would like to obtain is that the number of significant dimensions in the kernel $K(A,B)$ is small (This is true btw). For the sinc kernel, the eigenfunctions are prolate spheroidal wave functions, and it is known from, e.g., Slepian's work that only a few of them are sufficent to represent the Kernel well. Now, for the modified Kernel $K(A,B)$ I would like to obtain a similar result.

I have the following integral that I need to solve, but cannot make much progress. Anyone that have seen it before?

$$K(A,B) = \int_{-1}^1\int_{-1}^1 \cos(2\pi Ax)\cos(2\pi Bx)\mathrm{sinc}((x-y))(1-|x|)(1-|y|)\mathrm{d}x\mathrm{d}y.$$

The parameters A and B are less than 0.5 in magnitude.

In fact, the end result I would like to obtain is that the number of significant dimensions in the kernel $K(A,B)$ is small (This is true btw). For the sinc kernel, the eigenfunctions are prolate spheroidal wave functions, and it is known from, e.g., Slepian's work that only a few of them are sufficent to represent the Kernel well. Now, for the modified Kernel $K(A,B)$ I would like to obtain a similar result.

I have the following integral that I need to solve, but cannot make much progress. Anyone that have seen it before?

$$K(A,B) = \int_{-1}^1\int_{-1}^1 \cos(2\pi Ax)\cos(2\pi By)\mathrm{sinc}((x-y))(1-|x|)(1-|y|)\mathrm{d}x\mathrm{d}y.$$

The parameters A and B are less than 0.5 in magnitude.

In fact, the end result I would like to obtain is that the number of significant dimensions in the kernel $K(A,B)$ is small (This is true btw). For the sinc kernel, the eigenfunctions are prolate spheroidal wave functions, and it is known from, e.g., Slepian's work that only a few of them are sufficent to represent the Kernel well. Now, for the modified Kernel $K(A,B)$ I would like to obtain a similar result.

Source Link
Tommy Ding
Tommy Ding
  • 21
  • 2

integral over sinc-kernel

I have the following integral that I need to solve, but cannot make much progress. Anyone that have seen it before?

$$K(A,B) = \int_{-1}^1\int_{-1}^1 \cos(2\pi Ax)\cos(2\pi Bx)\mathrm{sinc}((x-y))(1-|x|)(1-|y|)\mathrm{d}x\mathrm{d}y.$$

The parameters A and B are less than 0.5 in magnitude.

In fact, the end result I would like to obtain is that the number of significant dimensions in the kernel $K(A,B)$ is small (This is true btw). For the sinc kernel, the eigenfunctions are prolate spheroidal wave functions, and it is known from, e.g., Slepian's work that only a few of them are sufficent to represent the Kernel well. Now, for the modified Kernel $K(A,B)$ I would like to obtain a similar result.