Fourier series is a representation of an arbitrary function with a specific period in the form of a series. In general terms, this solution is called the expansion of the element in an orthogonal basis. The expansion of functions in a Fourier series is a rather powerful tool for solving various problems due to the properties of this transformation during integration, differentiation, as well as shifting the expression by 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 "series" are and what they are for. Meanwhile, this transformation has entered our life quite tightly. It is used not only by mathematicians, but also by physicists, chemists, physicians, astronomers, seismologists, oceanographers, and many others. Let us take a closer look at the works of the great French scientist who made a 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 a sound wave. The oscillatory movements of elementary particles in an elastic medium are arranged in series (over the spectrum) of successive values of the volume level for tones of different heights. Further, the brain turns this data into sounds familiar to us. All this happens in addition to our desire or consciousness, in itself, but in order to understand these processes, it will take several years to study higher mathematics.
More on Fourier Transform
The Fourier transform can be carried out by analytical, numerical and other methods. Fourier series refer to the numerical method of decomposition of any oscillatory processes - from ocean tides and light waves to cycles of solar (and other astronomical objects) activity. Using these mathematical techniques, it is possible to analyze functions, representing any oscillatory processes as a series of sinusoidal components that go from minimum to maximum and vice versa. The Fourier transform is a function that describes the phase and amplitude of the sinusoids corresponding to a specific frequency. This process can be used to solve very complex equations that describe the dynamic processes that occur under the influence of thermal, light or electric energy. Also, the Fourier series makes it possible to isolate the constant components in complex vibrational signals, making it possible to correctly interpret the obtained experimental observations in medicine, chemistry, and astronomy.
Historical reference
The founding father of this theory is the French mathematician Jean Baptiste Joseph Fourier. His name was subsequently named this transformation. Initially, the scientist used his method to study and explain the mechanisms of thermal conductivity - the distribution of heat in solids. Fourier suggested that the initial irregular distribution of the heat wave can be decomposed into simple sinusoids, each of which will have its own temperature minimum and maximum, as well as its own phase. Moreover, each such component will be measured from minimum to 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 reduced the general distribution function, which is difficult to mathematically describe, to a very convenient to use series of periodic cosine and sine functions , in total giving the initial distribution.
The principle of transformation and the views of contemporaries
Contemporaries of the scientist - leading mathematicians of the early nineteenth century - did not accept this theory. The main objection was Fourier's assertion that a discontinuous function that describes a straight line or a discontinuous curve can be represented as a sum of sinusoidal expressions that are continuous. As an example, we can consider the “step” of Heaviside: its value is zero to the left of the gap and unity 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 faced a similar situation where a discontinuous expression would be described by a combination of continuous, ordinary functions, such as an exponent, a sinusoid, linear or quadratic.
What confused French mathematicians in Fourier theory?
After all, if the mathematician was right in his statements, then, summing up the infinite trigonometric Fourier series, you can get an accurate representation of the stepwise expression even if it has many similar steps. At the beginning of the 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 scope of thermal conductivity research. However, most scientists continued to be tormented by the question: "Can the sum of the sinusoidal series converge to the exact value of the discontinuous function?"
Fourier Series Convergence: An Example
The question of convergence arises whenever it is necessary to summarize infinite series of numbers. To understand this phenomenon, consider a classic example. Can you ever reach the wall if each subsequent step is half the size of the previous? Suppose you are two meters from the target, the first step approaches the halfway mark, the next step approaches 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 your intended goal in the strict mathematical sense. Using numerical calculations, we can prove that in the end you can approach an arbitrarily small given distance. This proof is equivalent to demonstrating that the total value of one second, one fourth, etc., will tend to unity.
Convergence Question: The Second Coming, or Lord Kelvin’s Appliance
This question was raised again at the end of the nineteenth century, when the Fourier series was tried to be used to predict the intensity of ebbs and flows. At this time, Lord Kelvin invented the device, which is an analog computing device that allowed sailors of the military and merchant fleet to track this natural phenomenon. This mechanism determined sets of phases and amplitudes from a table of tidal heights and their corresponding time points, carefully measured in this harbor during the year. Each parameter was a sinusoidal component of the expression of the tide height and was one of the regular components. The measurement results were entered into Lord Kelvin's computing device, synthesizing a curve that predicted the height of water as a temporary function for the next year. Very soon similar curves were compiled for all the harbors of the world.
And if the process is disrupted by a discontinuous function?
At that time, it seemed obvious that a tidal wave prediction device with a large number of counting elements could calculate a large number of phases and amplitudes and thus 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 data from the table of time moments is entered into the device, it calculates several Fourier coefficients. The original function is restored due to sinusoidal components (in accordance with the found coefficients). The discrepancy between the original and the restored expression can be measured at any point. When conducting repeated calculations and comparisons, it is clear that the value of the largest error does not decrease. However, they are localized in the region corresponding to the discontinuity point, and tend to zero at any other point. In 1899, this result was theoretically confirmed by Joshua Willard Gibbs of Yale.
Convergence of Fourier series and the development of mathematics in general
Fourier analysis is not applicable to expressions containing an infinite number of bursts in a certain interval. In general, the Fourier series, if the initial function is represented by the result of a real physical measurement, always converge. The convergence of this process for specific classes of functions has led to the emergence 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 basis was created for such expressions as the Dirac delta function (it describes a region of a single area concentrated in an infinitely small neighborhood of a point) and the Heaviside “step”. Thanks to this work, the Fourier series became applicable for solving equations and problems in which intuitive concepts appear: a point charge, a point mass, magnetic dipoles, as well as a concentrated load on a beam.
Fourier Method
Fourier series, in accordance with the principles of interference, begin with the decomposition of complex forms into simpler ones. For example, a change in the heat flux is explained by its passage through various obstacles from an insulating material of irregular shape or by a change in the surface of the earth - an earthquake, a change in the orbit of a celestial body - the influence of planets. As a rule, similar equations describing simple classical systems are elementarily solved for each individual wave. Fourier showed that simple solutions can also be summarized to obtain solutions to more complex problems. Expressed in the language of mathematics, Fourier series is a technique for representing an expression by the sum of harmonics - cosine and sine waves. Therefore, this analysis is also known as "harmonic analysis."
Fourier series - an ideal technique before the "computer era"
Before the creation of computer technology, the Fourier technique 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 can be directly applied by Newton's laws of 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 whole new level. This technique is firmly entrenched in almost all areas of science and technology. An example is digital audio and video. Its implementation became possible only thanks to the theory developed by the French mathematician in the early nineteenth century. Thus, the Fourier series in a complex form allowed a breakthrough in the study of outer space. In addition, this influenced 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 cases, the number of such expressions can be infinite. Moreover, the more their number is taken into account in the calculation, the more accurately the final result is obtained. Most often, the trigonometric functions of the cosine or sine are used as the simplest ones. In this case, the Fourier series are called trigonometric, and the solution of such expressions is called 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 main apparatus of the theory. In addition, it allows you to solve a number of problems in 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 laid the foundation for harmonic analysis.