About Higher Order Fourier Analysis

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This quastion is about the extension of Fourier analysis to higher orders. I know this is the approach to go beyond the linear phases. I am curious to understand it better. To be specific,

  1. Is it possible to briefly explain the higher order Fourier analysis with giving some clear mathematical representation of that analysis (just like the traditional form)?

  2. Would you please provide some applications of the higher order Forier analysis?

Thanks.