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Newton's "principles" opened the door to the study of natural mathematics, and colleagues in continental Europe extended Newton's ideas about the laws of nature to most fields of physical science. Following the wave equation (the wave equation vibrating from the violin has become one of the basic theories supporting modern science and technology), many other "gravity equations" have emerged one after another, such as electrostatic equation, elastic equation and heat flow equation.

Many equations are named after their inventors, such as Laplace equation and Poisson equation. This is not the case with the heat equation, which is both unimaginative and inaccurate. The heat equation was introduced by Joseph Fourier, whose ideas led to the birth of a new field of mathematics that branched far beyond its original source.

In 1807, Fourier submitted an article on heat flow to the French Academy of Sciences based on a new partial differential equation:

It is assumed that the metal rod is infinitely thin, the thermal diffusivity α is constant, and u (x, t) is the temperature at the x position and time t of the metal rod. So it should be called the temperature equation. He came up with a higher-dimensional version.

Where ▽ is Laplace operator.

It sure is

There are incredible similarities between the heat equation and the wave equation, but there is a key difference. Wave equation uses second-order time derivative

But in the heat equation, it is replaced by the first time derivative ∂ u / ∂ t. This change may seem small, but its physical significance is huge. Heat does not last indefinitely like the vibration of a violin string (assuming there is no friction or other damping according to the wave equation). On the contrary, with the passage of time, the heat dissipates unless there is a heat source that can heat it. So a typical question is: heating one end of a bar to keep its temperature stable and cooling the other end to achieve the same effect, how does the temperature change along the rod when the state of the metal bar stabilizes? The answer is that it is declining exponentially.

Another problem is how to determine the temperature variation with time after determining the initial temperature distribution along the metal bar. Maybe the left half starts with a higher temperature and the right half starts with a lower temperature. This equation tells us how heat spreads from the hot part to the cold part.

The heat equation is linear, so we can superimpose the solution. If the initial condition is

So the solution is

But initial conditions like this are a little unreal. To solve the problem I mentioned earlier, we need an initial condition where half of the metal bars have u (x, 0) = 1, and the other half is u (x, 0) = − 1. This initial condition is discontinuous and is called square wave in engineering terminology. But the sine and cosine curves are continuous. So the superposition of sine and cosine curves cannot represent square waves.

But what if infinite terms are allowed to be superimposed? We can try to express the initial conditions in the form of infinite series

Now it does seem possible to get a square wave. In fact, most coefficients can be set to zero, with only bounded terms where n is an odd number.

How to get square waves from sine and cosine. Left: sine wave component. Right: their sum and a square wave. The first few terms of the Fourier series are shown here. The additional term makes the approximation of the square wave more accurate.

Fourier even gives the general formulas of the coefficients axin and baun which represent the general condition f (x) in the form of integral.

After a long exploration of the expansion of the power series of trigonometric functions, he realized that there was a simpler way to derive these formulas. If you take two different trigonometric functions (such as cos2x and sin5x), multiply them, and integrate them from 0 to 2 pi, the result is zero. But if they are equal, assuming that they are all equal to sin5x, the integral of their product is not zero (it is pi). Suppose f (x) is the sum of a trigonometric series, and all the terms are multiplied by sin5x, and then integrated, and all the terms disappear, except for the term corresponding to sin5x, which is b_5sin5x. The integral here is pi. Divided by this, you get the Fourier formula of bau5, and the other coefficients are the same.

Fourier was severely criticized for not being rigorous enough, and Fourier was infuriated. His physical intuition told him that he was right. The real problem is that Euler and Bernoulli have been arguing for a long time about a similar problem in the wave equation, where heat diffuses exponentially over time and is replaced by infinite sinusoidal amplitudes. The basic mathematical problems are the same. In fact, Euler has published the integral formula of the coefficients in the wave equation.

However, Euler has never claimed that the formula applies to discontinuous functions, which is the most controversial aspect of Fourier research. The violin string model does not contain discontinuous initial conditions. But for heat, it is natural to consider keeping one area of a metal bar at one temperature and the adjacent area at another temperature. In practical application, the transition process is smooth and steep, but the use of discontinuous model is reasonable and convenient to calculate. In fact, the solution of the heat equation explains why the transition quickly becomes smooth and steep when the heat spreads to both sides.

Mathematicians began to realize that infinite series were "dangerous beasts". In the end, these complex problems were solved. In 1822, Fourier published his book, the Theory of Thermal Analysis.

We now know that while Fourier is right, his critics have good reason to worry about its rigor. Fourier analysis is good, but there are still some problems.

The question is, when does the Fourier series converge to the function it represents? In other words, if you take more and more items, will the approximate value of the function be better? Even Fourier knows that the answer is not always so. For example, at the midpoint of the temperature jump, the Fourier series of the square wave converges-but converges to the wrong value 0, but the value of the square wave is 1.

For most physical problems, it doesn't matter much to change the value of a function at an outlier. It's just slightly different at discontinuous points. For Fourier, such issues are not important. However, the convergence problem can not be ignored so rashly, because the discontinuity of the function may be much more complex than the square wave.

However, Fourier claims that his method applies to any function, so it should apply to functions where f (x) = 0 when x is rational and f (x) = 1 when x is irrational. This function is discontinuous everywhere. For such a function, at that time, the meaning of the integral was not clear. This is the real cause of the controversy. No one has ever defined what an integral is. No one has even defined what a function is. Even if you can close these loopholes, it's not just a question of whether the Fourier series converges or not. The real difficulty is to figure out in what sense it is convergent.

Solving these problems is tricky:

It requires a new integral theory, proposed by Henry Leberg.

Reconstruct the foundation of mathematics from the perspective of set theory, initiated by George Cantor.

He gained important insights from outstanding figures such as Riemann and used the abstract concepts of the 20th century to solve the problem of convergence.

The final conclusion is that, through the correct explanation, Fourier did solve the heat equation. But its real meaning is much broader. Apart from pure mathematics, the main beneficiaries are not thermodynamics, but engineering, especially electronic engineering.

In the most general form, the Fourier method represents a signal, determined by the function f. This is called the Fourier transform of waves. It uses spectrum instead of the original signal: this is a set of sine and cosine amplitudes and frequencies that encode the same information in different ways.

One application of this technology is the design of earthquake-resistant buildings. The Fourier transform of the vibration produced by a typical earthquake reveals the energy frequency of the earthquake. The building has its own natural vibration mode, and it will resonate with the earthquake, that is to say, the reaction is extremely strong. Therefore, the first step in building earthquake resistance is to ensure that the preferred frequency of the building is different from that of the earthquake. The frequency of earthquakes can be obtained by observation, while the frequency of buildings can be calculated by computer models.

This is just one of many aspects in which Fourier transform affects our lives "behind the scenes". Fourier transform has become a routine tool in science and engineering; its applications include removing noise from sound records; discovering the structure of large biochemical molecules such as DNA by x-ray diffraction; and improving radio reception and processing photos taken from the air. Here, I only focus on one of thousands of everyday applications: image processing.

Application of Fourier transform in graphic processing how does Fourier transform deal with pictures and does not affect the clarity of pictures?

The answer is data compression. Some of these processes are "lossless", which means that the original information can be retrieved from the compressed version if needed. This is necessary because most real-world images contain redundant information. For example, large chunks of sky are usually the same blue. You don't have to repeat the color and brightness information of blue pixels over and over again, so you can store the two diagonal coordinates of a rectangle and a few lines of short code (paint the whole area blue).

The human eye is not particularly sensitive to some features of the image, which can be recorded on a thicker scale without most people noticing. It is easy to compress information in this way, but it is irreversible (information loss).

Some cameras save the image as a JPEG file, which indicates that a specific data compression system is used. Software used to process and print photos, such as Photoshop, can decode the JPEG format and convert the data back to the picture. We often use JPEG files, and few people know that they are compressed, and fewer people want to know how this is done.

Fourier analysis has become a must for engineers and scientists, but for some purposes, it has a major drawback: sine and cosine have infinite terms. When Fourier's method tries to represent a compact signal, it encounters a problem. It requires a lot of sine and cosine to simulate a local spot of light. The problem is not to get the basic shape of the spot, but to make everything outside the spot equal to zero. What you need to do is to add more high-frequency sine and cosine to counteract unwanted signals. So the Fourier transform is useless for the light spot signal: the transformed signal is more complex than the original signal, and more data is needed to describe it.

Sine and cosine are chosen because they meet a simple condition. Formally, this means that they are orthogonal. Multiplying two basic sinusoidal waveforms and integrating them in a period can measure how close the relationship is between them. If this number is large, they are very similar; if it is zero (the condition of orthogonality), they are independent.

Fourier analysis is effective because its basic waveforms are orthogonal and complete, and if properly superimposed, they are sufficient to represent any signal. In fact, they provide a coordinate system in the space of all signals, just like a three-dimensional coordinate system in ordinary space. The main new feature is that there are now an infinite number of axes: each basic waveform has an axis. Once you get used to it, there will be no problem in math.

It is not difficult to find that there is a coordinate system different from Fourier in the infinite dimensional signal space. One of the most important discoveries in the whole field is a new coordinate system in which the basic waveform is limited to a limited space. They are called wavelets and can represent light spots very effectively because they are light spots. The light spot feature of wavelet makes it especially suitable for compressed images. One of their earliest large-scale practical applications is to store fingerprints. In addition, wavelet has many applications in medical imaging. In fact, wavelets are almost everywhere. Researchers in the fields of geophysics and electrical engineering have applied these technologies to their own fields.

This article comes from the official account of Wechat: Lao Hu Shuo Science (ID:LaohuSci). Author: I am Lao Hu.

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