## MATH7150 Fourier analysis课程简介

Harmonic analysis is a branch of mathematics that deals with the study of functions and signals in terms of their frequency components. It has many applications in areas such as signal processing, image analysis, differential equations, and probability theory. In this context, the term “harmonic” refers to functions that satisfy a certain differential equation called the Laplace equation, which is a central topic in harmonic analysis.

One of the main tools in harmonic analysis is the Fourier transform, which decomposes a function or signal into its frequency components. The Fourier transform is defined in terms of complex exponential functions, and it can be extended to functions on more general spaces, such as groups and manifolds.

## PREREQUISITES

Convergence of Fourier series is a fundamental topic in harmonic analysis. It concerns the question of under what conditions the Fourier series of a given function converges to the function itself. This question is intimately related to the properties of the underlying function, and it has many applications in areas such as partial differential equations, number theory, and probability theory.

Another important topic in harmonic analysis is the study of harmonic functions and their conjugates. Harmonic functions are solutions to the Laplace equation, and they have many interesting properties, such as the mean value property and the maximum principle. Conjugate harmonic functions are related to the concept of analytic functions, and they play an important role in complex analysis.

The Hilbert transform is a linear operator in harmonic analysis that is defined in terms of the Cauchy principal value of an integral. It has many applications in areas such as signal processing and image analysis.

Calderon-Zygmund theory is a modern branch of harmonic analysis that deals with the study of singular integrals, which are operators that involve the convolution of a function with a singular kernel. This theory has many applications in areas such as partial differential equations and image analysis.

## MATH7150 Fourier analysis HELP（EXAM HELP， ONLINE TUTOR）

Inverse Fourier transform. Prove that the inverse Fourier transform equals the Fourier transform, up to scaling. Concretely, show that for any $f:{-1,1}^n \rightarrow \mathbb{R}$ it holds that $\widehat{\widehat{f})}=2^{-n} f$.

To prove that the inverse Fourier transform equals the Fourier transform up to scaling, we need to show that for any function $f:\mathbb{R}^n\rightarrow\mathbb{C}$, we have $$f(x) = \frac{1}{(2\pi)^n}\int_{\mathbb{R}^n} \hat{f}(k)e^{ik\cdot x} dk.$$ Here, $k\cdot x$ denotes the inner product of the vectors $k$ and $x$, and $\hat{f}$ is the Fourier transform of $f$ defined by $$\hat{f}(k) = \int_{\mathbb{R}^n} f(x)e^{-ik\cdot x} dx.$$ We can prove this by starting with the definition of $\hat{f}$ and computing the inverse Fourier transform. We have \begin{align*} \widehat{\widehat{f}}(x) &= \int_{\mathbb{R}^n} \hat{f}(k)e^{ik\cdot x} dk\ &= \int_{\mathbb{R}^n} \left(\int_{\mathbb{R}^n} f(y)e^{-ik\cdot y} dy\right) e^{ik\cdot x} dk\ &= \int_{\mathbb{R}^n} f(y)\left(\int_{\mathbb{R}^n} e^{i(k\cdot x – k\cdot y)} dk\right) dy\ &= \int_{\mathbb{R}^n} f(y)\delta(x-y) dy\ &= f(x). \end{align*} Here, we have used the fact that the Fourier transform and the inverse Fourier transform are both linear operators, so we can interchange the order of integration and the Fourier transforms. We have also used the fact that the Fourier transform of the Dirac delta function $\delta(x)$ is $1$. Therefore, we have shown that $\widehat{\widehat{f}}=f$, which implies that $\widehat{\widehat{f}}=2^{-n}f$ since the Fourier transform is a unitary operator that preserves the norm of functions.

To prove the specific case for $f:{-1,1}^n \rightarrow \mathbb{R}$, we can use the definition of the Fourier transform on the discrete cube ${-1,1}^n$. We have \begin{align*} \hat{f}(k) &= \frac{1}{2^n}\sum_{x\in{-1,1}^n} f(x)e^{-ik\cdot x}\ &= \frac{1}{2^n}\sum_{x\in{-1,1}^n} f(x)\prod_{i=1}^n e^{-ik_i x_i}\ &= \frac{1}{2^n}\prod_{i=1}^n\left(\sum_{x_i\in{-1,1}}f(x_1,\ldots,x_n)e^{-ik_i x_i}\right)\ &= \frac{1}{2^n}\prod_{i=1}^n\left(f(1,0,\ldots,0)+f(-1,0,\ldots,0)e^{-ik_i}\right)\ &= \frac{1}{2^n}\sum_{S\subseteq [n]}(-1)^{|S|}\prod_{i\in S}k_i\left(\sum_{

1. Lower bounds for linearity testing. Let $A$ be a randomized algorithm, which has access to a boolean function $f: \mathbb{F}_2^n \rightarrow \mathbb{F}_2$, makes $q$ queries to $f$ (possibly adaptively), and satisfies that
• If $f$ is linear then $\operatorname{Pr}[A$ accepts $f]=1$.
• If $f$ is $\varepsilon$-far from linear then $\operatorname{Pr}[A$ accepts $f] \leq 1 / 2$.
Prove that $q \geq \Omega(1 / \varepsilon)$. That is, the BLR algorithm is optimal up to constants.

To prove the lower bound, we will construct a family of boolean functions $f_{\mathbf{u}}:\mathbb{F}_2^n\rightarrow\mathbb{F}2$, where $\mathbf{u}=(u_1,\dots,u_n)$ is a vector of signs $u_i\in{-1,1}$, such that any algorithm that makes at most $q$ queries to $f{\mathbf{u}}$ and distinguishes between the case where $\mathbf{u}$ is a vector of $1$s (which corresponds to a linear function) and the case where $\mathbf{u}$ is chosen uniformly at random from all possible vectors of signs will have error probability at least $1/2+\Omega(1/\sqrt{q})$. By Yao’s minimax principle, this will imply the desired lower bound.

We define $f_{\mathbf{u}}$ as follows: Let $X_1,\dots,X_n$ be independent random variables taking values in ${-1,1}$ with $\mathbb{P}(X_i=u_i)=\mathbb{P}(X_i=-u_i)=1/2$. We then define $f_{\mathbf{u}}(\mathbf{x})=\mathrm{sign}\left(\sum_{i=1}^n u_i x_i\right)$. In other words, $f_{\mathbf{u}}$ is the parity function with respect to the vector $\mathbf{u}$.

We now claim that the family ${f_{\mathbf{u}}}$ satisfies the desired properties. To see this, let $\mathbf{u}=(u_1,\dots,u_n)$ be a vector of signs, and let $\mathbf{v}$ be a uniformly random vector of signs. We then have the following two cases:

• Case 1: $\mathbf{u}=\mathbf{1}$. In this case, $f_{\mathbf{u}}$ is the linear function $f_{\mathbf{1}}(\mathbf{x})=\mathrm{sign}\left(\sum_{i=1}^n x_i\right)$, and so any algorithm that makes at most $q$ queries to $f_{\mathbf{u}}$ can be represented as a linear combination of the form $a_1 f_{\mathbf{u}1}+\dots+a_q f{\mathbf{u}q}$, where $\mathbf{u}1,\dots,\mathbf{u}q$ are vectors of signs and $a_1,\dots,a_q\in\mathbb{F}2$. Let $\mathbf{v}$ be a uniformly random vector of signs. Then we have \begin{align*} \mathbb{P}{\mathbf{x}\sim\mathbb{F}2^n}[a_1 f{\mathbf{u}1}(\mathbf{x})+\dots+a_q f{\mathbf{u}q}(\mathbf{x})\neq f{\mathbf{v}}(\mathbf{x})]&=\frac{1}{2^n}\sum{\mathbf{x}\in{-1,1}^n}(-1)^{a_1 f{\mathbf{u}1}(\mathbf{x})+\dots+a_q f{\mathbf{u}q}(\mathbf{x})+f{\mathbf{v}}(\mathbf{x})}\ &=\frac{1}{2^n}\sum{\mathbf{x}\in\

## Textbooks

• An Introduction to Stochastic Modeling, Fourth Edition by Pinsky and Karlin (freely
available through the university library here)
• Essentials of Stochastic Processes, Third Edition by Durrett (freely available through
the university library here)
To reiterate, the textbooks are freely available through the university library. Note that
you must be connected to the university Wi-Fi or VPN to access the ebooks from the library
links. Furthermore, the library links take some time to populate, so do not be alarmed if
the webpage looks bare for a few seconds.

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