内容来源：https://www.quantamagazine.org/what-is-the-fourier-transform-20250903/

内容总结：

【数学革命背后的声音：傅里叶变换如何重塑现代科学】
19世纪初，法国数学家约瑟夫·傅里叶在热传导研究中突破性提出：任何复杂函数均可分解为一系列基本波（频率）的叠加，这一技术后被命名为“傅里叶变换”。尽管最初因“描述尖锐温度跳变”的激进观点遭受质疑，如今该理论已成为数学、物理、工程等领域的基石。

傅里叶变换的核心原理类似于人耳对复合声波的解析——通过扫描所有可能频率，计算各频率对原始函数的贡献值，最终将复杂信号拆解为可量化的成分。这一方法不仅推动了“调和分析”数学分支的诞生，更广泛应用于文件压缩、音频增强、医学成像、引力波探测等领域。例如JPEG图像压缩即通过剔除高频细节信息实现数据精简，而量子力学中的不确定性原理亦可通过傅里叶变换数学框架阐释。

值得一提的是，1965年快速傅里叶变换（FFT）算法的出现极大提升了计算效率，使其成为信号处理的标配工具。数学家查尔斯·费弗曼评价：“若没有傅里叶变换，现代数学的很大一部分将不复存在。”从拿破仑时代的热力学研究到今天的数字化革命，傅里叶变换持续释放着跨越世纪的科学能量。

中文翻译：

什么是傅里叶变换？
引言
当我们聆听一段音乐时，耳朵正在进行精密的计算。长笛的高频颤音、小提琴的中音区与低音提琴的深沉嗡鸣，共同在空气中激荡起不同频率的压力波。当复合声波穿过耳道进入螺旋状的耳蜗后，不同长度的纤毛会对不同音高产生共振，将杂乱信号分解为若干基础音素的集合。

直到19世纪，数学家才掌握了这种运算方法。19世纪初，法国数学家让-巴普蒂斯·约瑟夫·傅里叶发现：任何函数都可以分解为一组基础波（即频率）的集合。将这些组成频率重新叠加，就能还原原始函数。这项被称为"傅里叶变换"的技术，让这位法国大革命的狂热支持者又掀起了一场数学革命。

由此发展出的谐波分析数学分支，专门研究函数的组成成分。很快数学家发现谐波分析与数论、微分方程、量子力学等数学物理领域存在深刻关联。如今傅里叶变换已应用于计算机领域，实现文件压缩、音频信号增强等功能。

"傅里叶分析对数学的影响难以估量，"纽约大学及弗拉铁非研究所的莱斯利·格林加德表示，"它几乎触及数学、物理、化学等所有领域。"

激情岁月
傅里叶1768年生于革命前夜的法国乱世。十岁成为孤儿的他在欧塞尔的修道院接受教育，此后十年一直在宗教与数学之间徘徊，最终放弃神职成为教师。他积极推动法国革命事业，却在1794年恐怖统治时期因反革命言论被捕入狱，面临断头台处决。

就在行刑前，恐怖统治宣告结束。1795年傅里叶重返数学讲台，数年后被任命为拿破仑·波拿巴的科学顾问随军远征埃及。在研究埃及文物期间，他开启了热传导数学研究。1801年法军撤离埃及前夕（罗塞塔石碑在此期间被英军缴获），傅里叶已完全沉浸于这项研究。

假设对金属棒一端加热，热量会扩散至整体温度均衡。傅里叶提出：热量在棒体内的分布可表示为简单波动的叠加。随着金属冷却，这些波动会因能量衰减逐渐平滑消失——高能高频波先衰减，低频波随后平息，宛如交响乐结束时各声部依次静默，从短笛到低音号渐次消逝。

这个激进理论在1807年巴黎学会会议上遭到著名数学家约瑟夫-路易·拉格朗日的质疑："这根本不可能实现"。最令同僚困惑的是极端案例：比如半冷半热的金属棒是否存在温度骤变？傅里叶坚持认为突变仍可用数学描述——只需叠加无限个简单曲线而非有限数量。当时多数数学家认为再多的平滑曲线也无法构成锐角。

如今我们知道傅里叶基本正确。"任何事物都可以表示为简单振荡的叠加，"普林斯顿大学数学家查尔斯·费弗曼解释道，"就像精心调校的音叉群能奏出贝多芬第九交响曲。"除非遇到最诡异的函数（无论放大多少倍都剧烈振荡者），否则这个理论始终成立。

那么傅里叶变换如何运作？

精妙的耳朵
执行傅里叶变换如同辨析香水的成分组合，或分辨复杂爵士和弦的构成音符。数学上，傅里叶变换是一个函数：它以复杂函数为输入，输出一组频率集合。写出对应频率的正余弦波并叠加，就能还原原始函数。

为实现这点，傅里叶变换会扫描所有可能频率，确定各频率对原函数的贡献度。以简单函数为例：当原函数与频率3的正弦波相乘时，显著波峰表明该频率存在贡献；而与频率5正弦波相乘时，波峰波谷相互抵消使均值趋零，说明该频率无贡献。

傅里叶变换对所有频率进行正余弦波乘积测试（实际运用复平面上的虚实数组合）。通过这种方式，复杂函数可被分解为少量数字，成为数学家的重要工具：难题经频率语言转换后往往迎刃而解。

若原函数存在锐利边缘（如数字信号中常见的方波），傅里叶变换会产生无限频率集合（即傅里叶级数）来无限逼近边缘。尽管早期数学家难以接受无限级数概念，如今它已成为函数分析的核心工具。

安可曲
傅里叶变换同样适用于图像等高维对象。灰度图像可视为二维函数（定义各像素亮度），变换将其分解为二维频率集合。这些频率定义的正余弦波形成不同方向的条纹图案，通过类似棋盘格的组合叠加可重构任何图像。

例如任何8x8图像都由64种基础模块组合而成。压缩算法可移除对应细节的高频信息而不影响视觉观感，这就是JPEG将复杂图像压缩为小体积数据的原理。

1960年代数学家詹姆斯·库利与约翰·图基提出快速傅里叶变换算法，极大提升了运算效率。此后所有信号处理领域都开始应用该技术。"它已成为日常生活的一部分，"格林加德表示。

这项技术被用于潮汐研究、引力波探测、雷达与核磁共振成像开发，能消除音频文件噪点并压缩存储各类数据。在量子力学领域，它甚至为不确定性原理提供数学基础：描述粒子位置概率的函数经傅里叶变换后，会显示其动量概率分布。当函数显示粒子大概率处于某位置（函数图形呈现尖峰）时，变换结果会非常分散，无法确定粒子动量，反之亦然。

傅里叶变换在纯数学领域也生根发芽。研究傅里叶变换及其逆变换重建原函数的谐波分析，成为研究波动的重要框架。数学家还发现谐波分析与数论存在深刻而意外的联系，借此探索整数关系（包括数学最大谜题之一的素数分布规律）。

"若没有傅里叶变换，不知多少数学分支将不复存在，"费弗曼感叹，"这个比例必定惊人。"

编者注：弗拉铁非研究所由西蒙斯基金会资助，该基金会同样资助本编辑独立杂志。西蒙斯基金会的资助决策不影响本刊报道内容。更多关于《量子杂志》与西蒙斯基金会关系的信息请参见此处。

英文来源：

What Is the Fourier Transform?
Introduction
As we listen to a piece of music, our ears perform a calculation. The high-pitched flutter of the flute, the middle tones of the violin, and the low hum of the double bass fill the air with pressure waves of many different frequencies. When the combined sound wave descends through the ear canal and into the spiral-shaped cochlea, hairs of different lengths resonate to the different pitches, separating the messy signal into buckets of elemental sounds.
It took mathematicians until the 19th century to master this same calculation.
In the early 1800s, the French mathematician Jean-Baptiste Joseph Fourier discovered a way to take any function and decompose it into a set of fundamental waves, or frequencies. Add these constituent frequencies back together, and you’ll get your original function. The technique, today called the Fourier transform, allowed the mathematician — previously an ardent proponent of the French revolution — to spur a mathematical revolution as well.
Out of the Fourier transform grew an entire field of mathematics, called harmonic analysis, which studies the components of functions. Soon enough, mathematicians began to discover deep connections between harmonic analysis and other areas of math and physics, from number theory to differential equations to quantum mechanics. You can also find the Fourier transform at work in your computer, allowing you to compress files, enhance audio signals and more.
“It’s hard to overestimate the influence of Fourier analysis in math,” said Leslie Greengard of New York University and the Flatiron Institute. “It touches almost every field of math and physics and chemistry and everything else.”
Flames of Passion
Fourier was born in 1768 amid the chaos of prerevolutionary France. Orphaned at 10 years old, he was educated at a convent in his hometown of Auxerre. He spent the next decade conflicted about whether to dedicate his life to religion or to math, eventually abandoning his religious training and becoming a teacher. He also promoted revolutionary efforts in France until, during the Reign of Terror in 1794, the 26-year-old was arrested and imprisoned for expressing beliefs that were considered anti-revolutionary. He was slated for the guillotine.
Before he could be executed, the Terror came to an end. And so, in 1795, he returned to teaching mathematics. A few years later, he was appointed as a scientific adviser to Napoleon Bonaparte and joined his army during the invasion of Egypt. It was there that Fourier, while also pursuing research into Egyptian antiquities, began the work that would lead him to develop his transform: He wanted to understand the mathematics of heat conduction. By the time he returned to France in 1801 — shortly before the French army was driven out of Egypt, the stolen Rosetta stone surrendered to the British — Fourier could think of nothing else.
If you heat one side of a metal rod, the heat will spread until the whole rod has the same temperature. Fourier argued that the distribution of heat through the rod could be written as a sum of simple waves. As the metal cools, these waves lose energy, causing them to smooth out and eventually disappear. The waves that oscillate more quickly — meaning they have more energy — decay first, followed eventually by the lower frequencies. It’s like a symphony that ends with each instrument fading to silence, from piccolos to tubas.
The proposal was radical. When Fourier presented it at a meeting of the Paris Institute in 1807, the renowned mathematician Joseph-Louis Lagrange reportedly declared the work “nothing short of impossible.”
What troubled his peers most were strange cases where the heat distribution might be sharply irregular — like a rod that is exactly half cold and half hot. Fourier maintained that the sudden jump in temperature could still be described mathematically: It would just require adding infinitely many simpler curves instead of a finite number. But most mathematicians at the time believed that no number of smooth curves could ever add up to a sharp corner.
Today, we know that Fourier was broadly right.
“You can represent anything as a sum of these very, very simple oscillations,” said Charles Fefferman, a mathematician at Princeton University. “It’s known that if you have a whole lot of tuning forks, and you set them perfectly, they can produce Beethoven’s Ninth Symphony.” The process only fails for the most bizarre functions, like those that oscillate wildly no matter how much you zoom in on them.
So how does the Fourier transform work?
A Well-Trained Ear
Performing a Fourier transform is akin to sniffing a perfume and distinguishing its list of ingredients, or hearing a complex jazzy chord and distinguishing its constituent notes.
Mathematically, the Fourier transform is a function. It takes a given function — which can look complicated — as its input. It then produces as its output a set of frequencies. If you write down the simple sine and cosine waves that have these frequencies, and then add them together, you’ll get the original function.
To achieve this, the Fourier transform essentially scans all possible frequencies and determines how much each contributes to the original function. Let’s look at a simple example.
Consider the following function:
The Fourier transform checks how much each frequency contributes to this original function. It does so by multiplying waves together. Here’s what happens if we multiply the original by a sine wave with a frequency of 3:
There are lots of large peaks, which means the frequency 3 contributes to the original function. The average height of the peaks reveals how large the contribution is.
Now let’s test if the frequency 5 is present. Here’s what you get when you multiply the original function by a sine wave with the frequency 5:
There are some large peaks but also large valleys. The new graph averages out to around zero. This indicates that the frequency 5 does not contribute to the original function.
The Fourier transform does this for all possible frequencies, multiplying the original function by both sine and cosine waves. (In practice, it runs this comparison on the complex plane, using a combination of real and imaginary numbers.)
In this way, the Fourier transform can decompose a complicated-looking function into just a few numbers. This has made it a crucial tool for mathematicians: If they are stumped by a problem, they can try transforming it. Often, the problem becomes much simpler when translated into the language of frequencies.
If the original function has a sharp edge, like the square wave below (which is often found in digital signals), the Fourier transform will produce an infinite set of frequencies that, when added together, approximate the edge as closely as possible. This infinite set is called the Fourier series, and — despite mathematicians’ early hesitation to accept such a thing — it is now an essential tool in the analysis of functions.
Encore
The Fourier transform also works on higher-dimensional objects such as images. You can think of a grayscale image as a two-dimensional function that tells you how bright each pixel is. The Fourier transform decomposes this function into a set of 2D frequencies. The sine and cosine waves defined by these frequencies form striped patterns oriented in different directions. These patterns — and simple combinations of them that resemble checkerboards — can be added together to re-create any image.
Any 8-by-8 image, for example, can be built from some combination of the 64 building blocks below. A compression algorithm can then remove high-frequency information, which corresponds to small details, without drastically changing how the image looks to the human eye. This is how JPEGs compress complex images into much smaller amounts of data.
In the 1960s, the mathematicians James Cooley and John Tukey came up with an algorithm that could perform a Fourier transform much more quickly — aptly called the fast Fourier transform. Since then, the Fourier transform has been implemented practically every time there is a signal to process. “It’s now a part of everyday life,” Greengard said.
It has been used to study the tides, to detect gravitational waves, and to develop radar and magnetic resonance imaging. It allows us to reduce noise in busy audio files, and to compress and store all sorts of data. In quantum mechanics — the physics of the very small — it even provides the mathematical foundation for the uncertainty principle, which says that it’s impossible to know the precise position and momentum of a particle at the same time. You can write down a function that describes a particle’s possible positions; the Fourier transform of that function will describe the particle’s possible momenta. When your function can tell you where a particle will be located with high probability — represented by a sharp peak in the graph of the function — the Fourier transform will be very spread out. It will be impossible to determine what the particle’s momentum should be. The opposite is also true.
The Fourier transform has spread its roots throughout pure mathematics research, too. Harmonic analysis — which studies the Fourier transform, as well as how to reverse it to rebuild the original function — is a powerful framework for studying waves. Mathematicians have also found that harmonic analysis has deep and unexpected connections to number theory. They’ve used these connections to explore relationships among the integers, including the distribution of prime numbers, one of the greatest mysteries in mathematics.
“If people didn’t know about the Fourier transform, I don’t know what percent of math would then disappear,” Fefferman said. “But it would be a big percent.”
Editor’s note: The Flatiron Institute is funded by the Simons Foundation, which also funds this editorially independent magazine. Simons Foundation funding decisions have no influence on our coverage. More information about the relationship between Quanta Magazine and the Simons Foundation is available here.