Fourier series is a representation of an arbitrarily taken function with a specific period as a series. In general terms, this solution is called the decomposition of an element in 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 this transformation when integrating, differentiating, as well as shifting an expression in an 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 these “rows” are and what they are for. Meanwhile, this transformation has become quite dense in our lives. It is used not only by mathematicians, but also by physicists, chemists, physicians, astronomers, seismologists, oceanographers and many others. Let's 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 the soundwaves. The oscillatory motions of elementary particles in an elastic medium are decomposed into rows (along the spectrum) of successive values of the volume level for tones of different heights. Next, the brain turns this data into sounds familiar to us. All this happens in addition to our desire or consciousness, by itself, but in order to understand these processes, it will take several years to study higher mathematics.
More about the Fourier Transform
Fourier transform can be carried out by analytical, numerical and other methods. Fourier series refer to the numeral way of decomposing 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 sinusoids corresponding to a specific frequency. This process can be used to solve very complex equations that describe dynamic processes that occur under the influence of thermal, light or electrical energy. Also, Fourier series make it possible to isolate the constant components in complex oscillatory signals, which made it possible to correctly interpret the obtained experimental observations in medicine, chemistry and astronomy.
Historical background
Founding father of this theoryJean Baptiste Joseph Fourier is a French mathematician. This transformation was subsequently named after him. Initially, the scientist applied his method to study and explain the mechanisms of heat conduction - the spread of heat in solids. Fourier suggested that the initial irregular distribution of a heat wave can be decomposed into the simplest sinusoids, each of which will have its own temperature minimum and maximum, as well as its own phase. In this case, 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, is called the Fourier transform of the temperature distribution expression. The author of the theory reduced the general distribution function, which is difficult to describe mathematically, to a very easy-to-handle series of periodic cosine and sine functions that add up to the original distribution.
The principle of transformation and the views of contemporaries
The scientist's contemporaries - the leading mathematicians of the early nineteenth century - did not accept this theory. The main objection was Fourier's assertion that a discontinuous function describing a straight line or a discontinuous curve can be represented as a sum of sinusoidal expressions that are continuous. As an example, consider Heaviside's "step": its value is zero to the left of the gap 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 had never encountered sucha situation where the discontinuous expression would be described by a combination of continuous, ordinary functions, such as exponential, 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 exact representation of the step expression even if it has many similar steps. At the beginning of the nineteenth century, such a statement seemed absurd. But despite all the doubts, many mathematicians have expanded the scope of the study of this phenomenon, taking it beyond the study of thermal conductivity. However, most scientists continued to agonize over the question: "Can the sum of a sinusoidal series converge to the exact value of a discontinuous function?"
Convergence of Fourier series: example
The question of convergence is raised whenever it is necessary to sum up infinite series of numbers. To understand this phenomenon, consider a classic example. Can you ever reach the wall if each successive step is half the size of the previous one? Suppose you are two meters from the goal, the first step brings you closer to the halfway point, the next one to the three-quarters mark, and after the fifth you will cover 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, one can prove that in the end one can get as close as one likes.small specified distance. This proof is equivalent to demonstrating that the sum value of one-half, one-fourth, etc. will tend to one.
Question of Convergence: The Second Coming, or Lord Kelvin's Appliance
Repeatedly this question was raised at the end of the nineteenth century, when Fourier series were tried to be used to predict the intensity of ebb and flow. At this time, Lord Kelvin invented a device, which 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 a table of tide heights and their corresponding time moments, carefully measured in a given harbor during the year. Each parameter was a sinusoidal component of the tide height expression and was one of the regular components. The results of the measurements were entered into Lord Kelvin's calculator, which synthesized a curve that predicted the height of the water as a function of time for the next year. Very soon similar curves were drawn up 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 tidal wave predictor 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 cases where the tidal expression, which followsto synthesize, contained a sharp jump, that is, it was discontinuous. In the event that data is entered into the device from the table of time moments, then it calculates several Fourier coefficients. The original function is restored thanks to the sinusoidal components (according to the found coefficients). The discrepancy between the original and restored expression can be measured at any point. When carrying out repeated calculations and comparisons, it can be seen 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 University.
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, Fourier series, if the original function is represented by the result of a real physical measurement, always converge. Questions of the convergence of this process for specific classes of functions have 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 an area of a single area concentrated in an infinitely small neighborhood of a point) and the Heaviside “step”. Thanks to this work, Fourier series became applicable tosolving equations and problems that involve intuitive concepts: point charge, 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 heat flow is explained by its passage through various obstacles made of irregularly shaped heat-insulating material or 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 summed to give solutions to more complex problems. Expressed in the language of mathematics, Fourier series is a technique for representing an expression as the sum of harmonics - cosine and sinusoids. Therefore, this analysis is also known as "harmonic analysis".
Fourier series - the ideal technique before the "computer age"
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 a complex form allows solving not only simple problems that can be directly applied to the laws of Newton's mechanics, but also fundamental equations. Most of the discoveries of Newtonian science in the nineteenth century were made possible only by Fourier's technique.
Fourier series today
With the development of Fourier transform computersraised to a whole new level. This technique is firmly entrenched in almost 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 a French mathematician at the beginning of the nineteenth century. Thus, the Fourier series in a complex form made it possible to make a breakthrough in the study of outer space. In addition, it influenced the study of the physics of semiconductor materials and plasma, microwave acoustics, oceanography, radar, seismology.
Trigonometric Fourier series
In mathematics, a 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 accurate the final result is. Most often, the trigonometric functions of 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 a means for the image, as well as the study of functions, it is the main 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, gave rise to 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, functionsreal variable, functional analysis, and also laid the foundation for harmonic analysis.
