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I'm not sure how you get your FFT image. If I apply a discrete Fourier transform to your image: img = Import["http://i.stack.imgur.com/lquy0.jpg"]; pixels = ImageData[img][[All, All, 2]]; pixels = pixels - Mean[Flatten@pixels]; (Setting the data to 0 mean removes the bright spot in the center of the FFT - the bin for frequency 0.) wnd1d = ...


The problem is that the command FourierCosSeries expects a function whose cosine series expansion can be evaluated exactly - while the function y2[theta1] is far too complicated. I would suggest replacing your final line of code with the numerical equivalent y2f[theta1_] := Evaluate[1/Pi*NIntegrate[y2[theta1], {theta1, 0, Pi}] + ...

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