Fourier series: history and influence of a mathematical mechanism on the development of science

The Fourier series is a representation of an arbitrarily taken function with a specific period in the form of a series. In general, this solution is called the expansion of an element along an orthogonal basis. The expansion of functions in a Fourier series is a fairly powerful tool for solving various problems due to the properties of the given transformation in the integration, differentiation, and also the shift of the expression for the argument and convolution.

A person who is not familiar with higher mathematics, as well as with the works of the French scientist Fourier, most likely will not understand what kind of "ranks" and what they are for. And yet this transformation has become quite dense in our life. It is used not only by mathematicians, but also by physicists, chemists, physicians, astronomers, seismologists, oceanographers and many others. Let us also get acquainted with the works of the great French scientist who made the discovery ahead of his time.

Man and the Fourier transform

Fourier series are one of the methods (along with analysis and others) of the Fourier transform. This process occurs every time a person hears a sound. Our ear automatically converts the sound wave into an automatic mode . Vibrational motions of elementary particles in an elastic medium are decomposed into series (according to the spectrum) of successive values of the loudness level for tones of different heights. Further, the brain turns these data into sounds that are familiar to us. All this happens in addition to our desire or consciousness, in and of itself, but in order to understand these processes, it will take several years to study higher mathematics.

More about Fourier transform

The Fourier transform can be carried out by analytical, numerative and other methods. The Fourier series refers to the numerical method of decomposition of any vibrational processes - from ocean tides and light waves to solar (and other astronomical objects) activity cycles. Using these mathematical techniques, you can parse the functions, representing any oscillatory processes as a series of sinusoidal components, which move from the minimum to the maximum and back. The Fourier transform is a function describing the phase and amplitude of the sinusoids corresponding to a certain frequency. This process can be used to solve very complicated equations that describe the dynamic processes that occur under the action of thermal, light or electric energy. Fourier series also make it possible to isolate constant components in complex vibrational signals, which makes it possible to correctly interpret the experimental observations obtained in medicine, chemistry, and astronomy.

Historical reference

The founding father of this theory is the French mathematician Jean Baptiste Joseph Fourier. His name was later called this transformation. Initially, the scientist used his method to study and explain the mechanisms of thermal conductivity - the propagation of heat in solids. Fourier suggested that the initial irregular distribution of the thermal wave can be decomposed into simple sinusoids, each of which will have its own temperature minimum and maximum, and also its phase. In this case, each such component will be measured from the minimum to the maximum and vice versa. The mathematical function that describes the upper and lower peaks of the curve, as well as the phase of each of the harmonics, was called the Fourier transform of the expression for the temperature distribution. The author of the theory has reduced the general distribution function, which is difficult to be mathematically described, to a series of periodic cosine and sine functions that are very convenient to handle, in the sum giving the initial distribution.

The principle of transformation and views of contemporaries

Contemporaries of the scientist - leading mathematicians of the early nineteenth century - did not accept this theory. The main objection was the Fourier assertion that a discontinuous function describing a straight line or a tearing curve can be represented as a sum of sinusoidal expressions that are continuous. As an example, we can consider Heiside's "step": its value is zero to the left of the discontinuity and one to the right. This function describes the dependence of the electric current on the time variable when the circuit is closed. Contemporaries of the theory at that time never encountered a similar situation, when a discontinuous expression would be described by a combination of continuous, ordinary functions, such as an exponential, a sinusoid, linear or quadratic.

What embarrassed the French mathematicians in the theory of Fourier?

After all, if a mathematician was right in his statements, then, summing up an infinite trigonometric Fourier series, one can get an accurate representation of the step expression even if it has many such steps. In the early nineteenth century, such a statement seemed absurd. But despite all doubts, many mathematicians have expanded the scope of the study of this phenomenon, taking it beyond the limits of research on heat conductivity. However, most scientists continued to suffer the question: "Can the sum of the sinusoidal series converge to the exact value of the discontinuous function?"

Convergence of Fourier series: an example

The question of convergence is raised every time it is necessary to add up infinite numbers of numbers. To understand this phenomenon, let us consider a classical example. Can you ever reach the wall if each subsequent step is half the previous one? Suppose that you are two meters from the goal, the first step brings you to the mark on the half way, the next - to the mark of three quarters, and after the fifth you will overcome almost 97 percent of the way. However, no matter how many steps you take, you will not achieve the intended goal in a strict mathematical sense. Using numerical calculations, it can be shown that in the end it is possible to approach an arbitrarily small predetermined distance. This proof is equivalent to demonstrating that the total value of one second, one fourth, etc., will tend to unity.

The question of convergence: the second coming, or Lord Kelvin's Device

Repeatedly this question was raised at the end of the nineteenth century, when the Fourier series was tried to apply the intensity of tides and tides to predict. At this time, Lord Kelvin invented a device that is an analog computing device that allowed sailors of the military and merchant fleet to track this natural phenomenon. This mechanism determined the sets of phases and amplitudes from the tide height table and the corresponding time points carefully measured in the harbor during the year. Each parameter was a sinusoidal component of the expression of the height of the tide and was one of the regular components. The results of the measurements were entered into a Kelvin calculating device, synthesizing a curve that predicted the water height as a temporary function for the next year. Very soon such curves were compiled for all the harbors of the world.

And if the process is broken by a discontinuous function?

At that time it seemed obvious that a device predicting a tidal wave with a large number of counting elements could calculate a large number of phases and amplitudes and so provide more accurate predictions. Nevertheless, it turned out that this regularity is not observed in those cases when the tidal expression, which should be synthesized, contained a sharp jump, that is, it was discontinuous. In the event that the device enters data from the table of time moments, it calculates several Fourier coefficients. The original function is restored due to sinusoidal components (in accordance with the coefficients found). The discrepancy between the original and the restored expression can be measured at any point. When performing repeated calculations and comparisons, it is clear that the value of the largest error is not reduced. However, they are localized in the region corresponding to the point of discontinuity, and at any other point they tend to zero. In 1899, this result was theoretically confirmed by Joshua Willard Gibbs of Yale University.

The convergence of Fourier series and the development of mathematics in general

The Fourier analysis is not applicable to expressions containing an infinite number of bursts on a certain interval. In general, the Fourier series, if the original function is represented by the result of a real physical dimension, always converge. Questions of the convergence of this process for specific classes of functions led to the appearance of new sections in mathematics, for example, the theory of generalized functions. It is associated with such names as L. Schwartz, J. Mikusinsky and J. Temple. Within the framework of this theory, a clear and precise theoretical framework was created for such expressions as the Dirac delta function (it describes the area of a single area concentrated in an infinitesimal neighborhood of a point) and the Heaviside's "step". Due to this work, the Fourier series became applicable to solving equations and problems in which intuitive concepts appear: point charge, point mass, magnetic dipoles, and also concentrated load on the beam.

The Fourier method

The Fourier series, in accordance with the principles of interference, begin with the decomposition of complex forms into simpler ones. For example, the change in the heat flow is due to its passage through various obstacles from heat-insulating material of irregular shape or by changing the earth's surface-an earthquake, a change in the orbit of the celestial body, by the influence of planets. As a rule, similar equations describing simple classical systems are solved elementary for each individual wave. Fourier showed that simple solutions can also be summed up to solve more complicated problems. Expressed in the language of mathematics, the Fourier series is a technique for expressing an expression by the sum of harmonics - cosine and sinusoid. Therefore, this analysis is also known as "harmonic analysis".

The Fourier series is an ideal technique before the "computer age"

Before the creation of computer technology, Fourier's method was the best weapon in the arsenal of scientists when working with the wave nature of our world. The Fourier series in complex form allows us to solve not only simple problems that are amenable to the direct application of the laws of Newtonian mechanics, but also fundamental equations. Most of the discoveries of Newtonian science of the nineteenth century became possible only thanks to the Fourier method.

Fourier series today

With the development of computers, Fourier transforms have risen to a qualitatively new level. This technique is firmly entrenched in virtually all areas of science and technology. An example is a digital audio and video signal. Its realization became possible only thanks to the theory developed by the French mathematician in the early nineteenth century. Thus, the Fourier series in complex form allowed to make a breakthrough in the study of outer space. In addition, this affected the study of the physics of semiconductor materials and plasma, microwave acoustics, oceanography, radar, seismology.

Trigonometric Fourier series

In mathematics, the Fourier series is a way of representing arbitrary complex functions as a sum of simpler ones. In general, the number of such expressions can be infinite. In this case, the more their number is taken into account in the calculation, the more accurately the final result is obtained. Most often, trigonometric cosine or sinus functions are used as the simplest ones. In this case, the Fourier series is called trigonometric, and the solution of such expressions is the expansion of the harmonic. This method plays an important role in mathematics. First of all, the trigonometric series provides the means for the image, as well as the study of functions, it is the basic apparatus of the theory. In addition, it allows solving a number of problems of mathematical physics. Finally, this theory contributed to the development of mathematical analysis, brought to life a number of very important sections of mathematical science (the theory of integrals, the theory of periodic functions). In addition, it served as a starting point for the development of the following theories: sets, functions of a real variable, functional analysis, and also initiated a harmonic analysis.