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Our approach to higher order Fourier analysis is to study the ultra productof finite (or compact) Abelian groups on which a new algebraic theory appears.This theory has consequences on finite (or compact) groups usually in the formof approximative statements. The present paper is the second part of a paper inwhich higher order characters and decompositions were introduced. We generalizethe concept of the Pontrjagin dual group and introduce higher order versions ofit. We study the algebraic structure of the higher order dual groups. We provea simple formula for the Gowers uniformity norms in terms of higher orderdecompositions. We present a simple spectral algorithm to produce higher orderdecompositions. We briefly study a multi linear version of Fourier analysis.Along these lines we obtain new inverse theorems for Gowers's norms.

Higher order Fourier analysis as an algebraic theory II