作为代数理论的高阶傅里叶分析III
Higher order Fourier analysis as an algebraic theory III
摘要：
对于每一个自然数k，我们引入了阿贝尔群上函数的k阶卷积的概念。研究了函数代数在极限下的保卷积自同构群。事实证明，这样的群有k-幂零因子群，解释了为什么k阶傅里叶分析具有非交换特征。为了证明我们的方法在二次的情况下，我们开发了一个新的二次表示论有限阿贝尔群。本文将阿贝尔群的四次诣零态射的概念引入到两步诣零流形中。我们证明了一个结构定理，即有限阿贝尔群上的任何有界函数都可分解为一个结构部分（它是具有有界复杂度连续函数的诣零态射的合成）和一个具有小U3范数的随机部分。它蕴涵了一个新的U3范数逆定理。(The Un，n>3的一般情况将在本序列的下一部分中讨论。我们指出，我们的框架创建有趣的极限对象的功能有限（或紧凑）阿贝尔群，可测函数的零流形。
英文摘要：
For every natural number k we introduce the notion of k-th order convolution of functions on abelian groups. We study the group of convolution preserving automorphisms of function algebras in the limit. It turns out that such groups have k-nilpotent factor groups explaining why k-th order Fourier analysis has non-commutative features. To demonstrate our method in the quadratic case we develop a new quadratic representation theory on finite abelian groups. We introduce the notion of a quadrtic nil-morphism of an abelian group into a two step nil-manifold. We prove a structure theorem saying that any bounded function on a finite abelian group is decomposable into a structured part (which is the composition of a nil-morphism with a bounded complexity continuous function) and a random looking part with small U3 norm. It implies a new inverse theorem for the U3 norm. (The general case for Un, n>3 will be discussed in the next part of this sequence.) We point out that our framework creates interesting limit objects for functions on finite (or compact) abelian groups that are measurable functions on nil-manifolds.
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