7.1 Introduction to Fourier analysis on finite abelian groups

Fourier analysis on finite abelian groups is a powerful tool in additive combinatorics. It transforms complex functions into simpler ones, making it easier to study group structures and solve problems.

This intro covers the basics of Fourier transforms, their properties, and applications. We'll look at how to compute them, inversion formulas, and key theorems that make Fourier analysis so useful in combinatorics.

Fourier Transform on Finite Abelian Groups

Finite Abelian Groups and Fourier Transform

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• A finite abelian group is a group with a finite number of elements where the group operation is commutative
• The Fourier transform on a finite abelian group $G$ is a linear map from the space of complex-valued functions on $G$ to itself, denoted by $\hat{f}(\chi) = \sum_{x \in G} f(x)\chi(x)$, where $\chi$ is a character of $G$
  • Characters are homomorphisms from the group to the complex unit circle
  • Example: For the cyclic group $\mathbb{Z}/n\mathbb{Z}$, the characters are given by $\chi_k(x) = e^{2\pi ikx/n}$, where $k = 0, 1, \ldots, n-1$
• The Fourier transform is an isomorphism between the space of complex-valued functions on $G$ and the space of complex-valued functions on the dual group $\hat{G}$, which consists of all characters of $G$

Properties of the Fourier Transform

• The Fourier transform is a unitary operator, meaning that it preserves the inner product of functions
  • This property is crucial for the development of Plancherel's theorem
• The Fourier transform of a real-valued function is conjugate symmetric, i.e., $\hat{f}(\chi) = \hat{f}(\chi^{-1})^*$
  • This property can be used to simplify computations and reduce memory requirements when working with real-valued functions
• The Fourier transform of a convolution of two functions is the pointwise product of their Fourier transforms, and vice versa (convolution theorem)
  • This property allows for the study of the behavior of convolutions in the Fourier domain, which can simplify certain computations and provide insights into the structure of the convolution

Computing the Fourier Transform

Computation of the Fourier Transform

• To compute the Fourier transform of a function $f$ on a finite abelian group $G$, one needs to evaluate the sum $\hat{f}(\chi) = \sum_{x \in G} f(x)\chi(x)$ for each character $\chi$ of $G$
  • This computation can be done directly using the definition of the Fourier transform
  • Example: For the group $\mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$, the Fourier transform of a function $f$ can be computed by evaluating the sum for each of the four characters
• For the direct product of cyclic groups $(\mathbb{Z}/n_1\mathbb{Z}) \times \ldots \times (\mathbb{Z}/n_d\mathbb{Z})$, the characters are given by the products of the characters of the individual cyclic groups
  • This property allows for the computation of the Fourier transform on direct product groups using the Fourier transforms on the individual cyclic groups

Fast Fourier Transform (FFT)

• The Fourier transform can be computed efficiently using the Fast Fourier Transform (FFT) algorithm, which reduces the computational complexity from $O(|G|^2)$ to $O(|G| \log |G|)$
  • The FFT algorithm exploits the symmetries and periodicities of the characters to reduce the number of computations required
  • Example: The Cooley-Tukey FFT algorithm recursively divides the computation into smaller subproblems, which can be solved efficiently and combined to obtain the final result
• The FFT algorithm is widely used in various applications, such as signal processing, data compression, and polynomial multiplication
  • The efficiency of the FFT algorithm has made it a fundamental tool in many areas of science and engineering

Fourier Inversion and Plancherel's Theorem

Fourier Inversion Formula

• The Fourier inversion formula states that a function $f$ can be recovered from its Fourier transform $\hat{f}$ by the formula $f(x) = \frac{1}{|G|} \sum_{\chi \in \hat{G}} \hat{f}(\chi)\chi(x^{-1})$
  • This formula allows for the reconstruction of a function from its Fourier transform
  • The Fourier inversion formula is a consequence of the fact that the Fourier transform is an isomorphism between the space of functions on $G$ and the space of functions on $\hat{G}$
• The Fourier inversion formula is a powerful tool for analyzing functions on finite abelian groups and their Fourier transforms
  • It can be used to solve equations involving functions and their Fourier transforms
  • It also provides a way to understand the relationship between a function and its Fourier transform

Plancherel's Theorem

• Plancherel's theorem states that the Fourier transform preserves the $L^2$ norm of functions, i.e., $\sum_{x \in G} |f(x)|^2 = \frac{1}{|G|} \sum_{\chi \in \hat{G}} |\hat{f}(\chi)|^2$
  • This theorem is a consequence of the fact that the Fourier transform is a unitary operator on the space of complex-valued functions on $G$
  • Plancherel's theorem is an important tool in the study of the properties of the Fourier transform and its applications
• Plancherel's theorem has several important implications and applications
  • It allows for the study of the energy distribution of a function in the Fourier domain
  • It is used in the proof of the uncertainty principle for finite abelian groups, which states that a function and its Fourier transform cannot both be highly concentrated
  • It is also used in the development of sampling theorems and the study of signal processing on finite abelian groups

Convolution Theorem and Applications

Convolution and Convolution Theorem

• The convolution of two functions $f$ and $g$ on a finite abelian group $G$ is defined as $(f * g)(x) = \sum_{y \in G} f(y)g(x-y)$
  • Convolution is a mathematical operation that combines two functions to produce a third function
  • Example: For functions $f$ and $g$ on the group $\mathbb{Z}/n\mathbb{Z}$, the convolution is given by $(f * g)(x) = \sum_{y=0}^{n-1} f(y)g(x-y)$
• The convolution theorem states that the Fourier transform of a convolution is the pointwise product of the Fourier transforms, i.e., $(\hat{f} * \hat{g})(\chi) = \hat{f}(\chi)\hat{g}(\chi)$
  • This theorem allows for the study of the behavior of convolutions in the Fourier domain, which can simplify certain computations and provide insights into the structure of the convolution
  • The convolution theorem is a consequence of the fact that the Fourier transform is a homomorphism from the convolution algebra to the pointwise multiplication algebra

Applications in Additive Combinatorics

• In additive combinatorics, the convolution theorem is used to analyze the additive structure of sets and the behavior of sum sets, difference sets, and other combinatorial objects
  • Sum sets and difference sets are important objects in additive combinatorics that capture the additive structure of sets
  • Example: For sets $A$ and $B$ in a finite abelian group $G$, the sum set is defined as $A + B = \{a + b : a \in A, b \in B\}$
• The convolution theorem can be used to prove results such as the Cauchy-Davenport theorem, which bounds the size of the sum set of two subsets of a finite abelian group
  • The Cauchy-Davenport theorem states that for subsets $A$ and $B$ of $\mathbb{Z}/p\mathbb{Z}$, where $p$ is prime, $|A + B| \geq \min\{p, |A| + |B| - 1\}$
  • The proof of this theorem relies on the properties of the Fourier transform and the convolution theorem
• The convolution theorem is also used in the study of exponential sums and character sums, which are important tools in additive combinatorics and number theory
  • Exponential sums and character sums are sums of the form $\sum_{x \in G} f(x)\chi(x)$, where $f$ is a function and $\chi$ is a character
  • The convolution theorem can be used to estimate the size of these sums and to study their cancellation properties