“In summary the proposed research appears to be valid and
has applications. It is guaranteed to produce doctoral dissertations."
    -An expert researcher in the field in his review of this research for a US federal research funding agency.
 
"... Congratulations. It does seem that you have a novel and powerful method for solving integral equations. ... .
I admire what appears to be a brand new and promisingly important advancement to spatial signal processing."   
      --A Distinguished Professor of Engineering in his comments on this research.

"In Mathematics, sometimes, the simplest results are the most useful results."
       -- A Professor of Mathematics in his comments on this research.

COMPLETE  EXPERT  REVIEW  COMMENTS  ON  THIS  RESEARCH  RESULTS:

A research proposal was submitted to a U.S. Federal Research Funding Agency seeking funds for doing further research on RTs. Technical comments of an expert reviewer of the proposal are included below.

Technical Review of Proposal 50608-CI: "Rao Transforms:  A New Theoretical and Computational Framework for Linear and Non-linear Integral Equations."
February, 2007.

Overall:
In summary the proposed research appears to be valid and has applications. It is guaranteed to produce doctoral dissertations.

Potential of the proposal to achieve the stated objective of the research:
The proposal aims at developing techniques to solve linear and non-linear integral equations. The PI has isolated a method called Rao Transforms that he proposes to use to solve integral equations. The technique has been proved by the author to be more effective than conventional SVD method in handling image restoration from blurring. The technique is also more suited for implementing on parallel and distributed processing architectures. The author also demonstrates its effectiveness in solving various more general integral equations viz. Fredholm, Volterra, Uryshon, etc., of which image restoration from blurring is a specific case of Fredholm.

Likelihood of the proposed method to develop new capabilities or enhance existing capabilities:
The method of Rao Transforms has been proven to have computational advantages (polynomial speedup) over conventional SVD method in specific applications and is shown to perform as well as SVD in the worst case scenario. The author also compares RT method to Fast Fourier Transform (FFT) to highlight minor improvements in the context of image restoration. Furthermore the proposed method has been shown to have applications in solving more general class of Linear and non-linear differential equations of various important types, such as, Laplace, Poisson, Helmholtz, and Boundary value problems of certain elliptic type PDE's by recasting them as integral equations. However the specific advantages over conventional methods itself has been left for future research. Given, its advantages in solving some types of problems it is likely that the benefits will carry over to solving the similar class of more general problems of both differential and integral types.

Overall potential of the proposed effort:
If certain class of differential equations and boundary value problems could be solved more effectively, it is likely to have application in problems of computational and finite element fluid mechanics. Specifically could help in the hull design of underwater vehicles. Solving integral equation will also likely impact computational physics.