Major Problem with FT
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A personal working note by Phil dated 8.2.13, written for his Spectral Theory book project. He concludes that everything from Section 32 onward uses an unneeded "ensemble" and averaging, since P(ω) for any pulse train can be computed directly with a simple method using R''(z). He plans to start a "no ensemble FT update" folder and revise sections one at a time, and doubts he can meet his Aug 12 departure deadline.
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Major Problem with FT PhL 8.2.13
Well, this was my Big Nightmare, and it has finally soaked in that I have a huge problem. Xiong I guess was the last straw. Everything I have done starting from Section 32, although not strictly WRONG, is done in a completely stupid way which makes the whole paper stupid. I am now of the feeling that there is no need for any kind of "ensemble" in this document, so all those <...> averages are unnecessary. Using a very simple method involving R"(z), you can compute P(ω) for any pulse train with no need for ensembles. This then relieves me of the task of jumping through hoops in order to create an ensemble with the right properties! I must admit, I have spent many days trying to "tune" this ensemble. All that time is wasted in retrospect, and I doubt now I can meet my Aug 12 Cod departure deadline. This problem is a LARGE problem and permeates everything after Section 32 which is a lot of the doc.
So I will now start a new folder called "no ensemble FT update" and start editing sections one at a time and see if I run into trouble.