On the Inverse Ultrahyperbolic Klein-Gordon Kernel

Round 1

Reviewer 1 Report

In line 14 of page 2, revise the sentence "for some advance in ..."

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This manuscript is quite similar to the paper "On the convolution and inversion of diamond Klein Gordon kernel", published on International Journal of Pure and Applied Mathematics (IJPAM), Volume 83, No. 2, 2013, 331-348. The definition of the operator W_{alpha} is only slightly different from the operator T_{alpha} defined in the paper above, but the main theorem in section 4 is exactly the same as Theorem 5.1 in the paper published on IJPAM. 

Author Response

Dear the reviewer,   I thank you for your careful reading of the manuscript and helpful    comments and suggestions. I have made revisions according to your    comments and suggestions, described in attached file.         Best regards,   Dr.Kamsing Nonlaopon

Author Response File: Author Response.pdf

Reviewer 2 Report

This paper is dealt with the convolution of W_alpha, the ultra-hyperbolic Klein-Gordon kernel, and this content seems to be reasonable. As the above reason, I would like to recommend this manuscript as 'unaltered publication'. Thanks, and have a good day. 

Author Response

Dear the reviewer,We thank you for your careful reading of the manuscript and helpful comments  and suggestions.      Best regards, Dr.Kamsing Nonlaopon

Author Response File: Author Response.pdf

Reviewer 3 Report

Dear author, please add the conclusion section.

Also, the list of references does not contain articles for the last 5 years. Fix it, please.

Author Response

Dear the reviewer,We thank you for your careful reading of the manuscript and helpful comments  and suggestions. We have made revisions according to your comments and  suggestions, as described in attached file.     Best regards, Dr.Kamsing Nonlaopon

Author Response File: Author Response.pdf

Round 2

Reviewer 1 Report

The definition of diamond Klein-Gordon kernel T_a and ultra hyperbolic Klein-Gordon kernel W_a are quite similar. The only difference is that the kernel T_a is defined using convolution with S_{a+2r}. If you take S_{a+2r} to be a constant function, the kernel T_a will be exactly the same as W_a. In other words, the current manuscript is discussing a very simple special case of another paper.

Moreover, these two operators share many common properties. For example, equations (45)-(48) are exactly the same as equations (4.1)-(4.4) in the paper "On the convolution and inversion of diamond Klein-Gordon kernel" published on International Journal of Pure and Applied Mathematics (IJPAM), Volume 83, No. 2, 2013, 331-348. The main theorem on page 9 of the manuscript is exactly the same as the main theorem in the published paper I mentioned above. That's not surprising because the current manuscript is only discussing a simple special case.