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I must find a kernel that statisfies as follows: enter image description here

In the my reference paper, the author suggest gaussian kernel that is enter image description here

The purpose of that kernel is that it will take a weight for each points around center point (mean value), ànd that weighted value decreases drastically to zero as points go away from center point

The gaussian kernel is very good to approximate that properties. But selectec the kernel size is very challenge task. Could you suggest to me the other kernel that can satisfy above properties

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You can use any unimodal, symmetric, univariate, continuous density instead of the normal distribution (e.g. a Cauchy density). All of these densities satisfy conditions 1-3. Usually some kernels are better in terms of certain criterion but worst in terms of some other, there is no Panacea for the choice of a kernel. In practice, with moderate samples, all of them perform similarly. The crucial choice is the bandwidth parameter ($\sigma$, in you case). Take a look at the wikipedia article of KDE for a discussion on this point, other kernels, as well as relevant references:

http://en.wikipedia.org/wiki/Kernel_density_estimation

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