Fourier series - a representation of an arbitrarily chosen function to a specific period in a row.In general terms, the decision referred to the expansion element of the orthogonal basis.The expansion of functions in Fourier series is a pretty powerful tool in solving various problems due to the properties of the transformation in the integration, differentiation, and shift the argument expressions and convolution.
person who is not familiar with higher mathematics, as well as with the work of the French scientist Fourier likely will not understand what the "ranks" and what they do.Yet this transformation is quite firmly entered our lives.It is used not only mathematics, but also physicists, chemists, doctors, astronomers, seismologists, oceanographers and others.Let us, and we take a closer look at the works of the great French scientist who made the discovery, ahead of his time.
Man and the Fourier transform
Fourier series is one of the methods (along with analysis and others) of the Fourier transform.This process occurs every time a person hears a sound.Our ears automatically converts the sound wave.The vibrational motion of elementary particles in an elastic medium are arranged in series (in the spectrum) consecutive volume level for the tones of different pitches.Next, the brain converts the data into sounds familiar to us.All this comes in addition to our desire or consciousness itself, but in order to understand these processes will take several years to study higher mathematics.
details about the Fourier transform
Fourier transform can be performed analytical, numerals and other methods.Fourier series are numeral process for decomposing any oscillatory processes - from ocean tides and waves of light to solar cycles (and other astronomical objects) activity.Using these mathematical techniques can disassemble functions representing any oscillatory processes in a number of sinusoidal components that go from minimum to maximum and back.The Fourier transform is a function describing the phase and amplitude of sinusoids corresponding to a particular frequency.This process can be used to address the very complex equations that describe the dynamic processes occurring under the action of heat, light or electric energy.Also, the Fourier series used to distinguish DC components in complex waveforms, making it possible to correctly interpret the experimental observations in medicine, chemistry and astronomy.
Background
founding father of this theory is the French mathematician Jean Baptiste Joseph Fourier.His name was subsequently called this transformation.Initially, the researchers used a technique to study and explain the mechanisms of heat conduction - the heat propagation in solids.Fourier assumed that the initial distribution of irregular heat wave can be decomposed into simple sinusoid, each of which will have its temperature minimum and maximum, as well as its phase.Thus each such component to be measured from minimum to maximum and vice versa.The mathematical function that describes the upper and lower peaks of the curve, and the phase of each harmonic, called the Fourier transform of the expression of the temperature distribution.The author of the theory of reduced overall distribution function, which is difficult to mathematical description, in a very easy to handle a number of periodic functions of sine and cosine, giving a total of the initial distribution.
principle of conversion and the views of contemporaries
contemporaries scientist - the leading mathematicians of the early nineteenth century - did not accept this theory.The main objection was the approval of Fourier that breaking function describing a straight line or curve is torn, it can be represented as a sum of sinusoidal expressions that are continuous.As an example, consider the "step" Heaviside: its value is zero to the left of the gap and the right unit.This function describes the dependence of the electric current from the temporary variable for the closure of the circuit.Contemporaries theory at that time never encountered a similar situation when breaking expression describes to a combination of continuous, common functions, such as exponential, sine, linear or quadratic.
that confuses the French mathematicians in the theory of Fourier?
After all, if a mathematician was correct in his assertions, then, summing an infinite trigonometric Fourier series, you can get an accurate representation of the step of expression, even if it has many similar steps.In the early nineteenth century, this statement seemed absurd.But despite all the doubts, many mathematicians have expanded the scope of the study of this phenomenon, moving it beyond the research of thermal conductivity.However, most scientists continued to suffer the question: "Can the sum of sine series converges to the exact value of discontinuous function?"
Convergence of Fourier series: the example
issue of convergence raised whenever necessary summation of infinite series of numbers.To understand this phenomenon, consider the classic example.Could you ever reach the wall when each step will be half the previous?Suppose you are two meters from the goal, the first step closer to the halfway mark, the next - to the level of three-quarters, and after the fifth you overcome almost 97 percent of the way.However, no matter how many steps you make, the intended target you reach in the strict mathematical sense.Using numerical calculations, we can prove that in the end can be approached on an arbitrarily small given distance.This is equivalent to a proof demonstrating that the total value of one half, one fourth, and so on. E. Will tend to unity.
question of convergence: the second coming, or Device Lord Kelvin
again the question arose in the late nineteenth century, when the Fourier tried to use to predict the intensity of ebbs and flows.At that time, Lord Kelvin was invented device is an analog computing device which allows mariners military and merchant navy to track this natural phenomenon.This mechanism defines a set of phases and amplitudes of the table height of the tides and the corresponding time moments, carefully measured in the harbor during the year.Each parameter is a sinusoidal component tide of expression is one of the regular components.The measurement results are input to the computing device Lord Kelvin, synthesizing curve, which predicts the height of water as time function for the next year.Very soon these curves were made for all the harbors of the world.
And if the process will be broken discontinuous function?
At the time it seemed obvious that the device predicting a tidal wave, with lots of elements accounts can calculate a large number of phases and amplitudes, and so provide a more accurate prediction.However, it turned out that this pattern is not observed in cases where the tidal expression that will be synthesized, contained a sharp jump, ie it is discontinuous.In that case, if data is entered into the device from a table of time points, it calculates few Fourier coefficients.The original function is restored thanks to the sinusoidal component (in accordance with the found coefficients).The discrepancy between the original and the reconstructed expression can be measured at any point.During the repeated computation and comparison shows that the value of the greatest error is reduced.However, they are localized in the region corresponding to the point of rupture, and any other points tends to zero.In 1899, this result was confirmed theoretically Joshua Willard Gibbs of Yale University.
Convergence of Fourier series and the development of mathematics in general
Fourier analysis does not apply to expressions containing an infinite number of bursts at a certain interval.In general Fourier series, if the original function of presenting the results of the actual physical measurement always converge.Questions of convergence of the process for specific classes of functions have led to new branches of mathematics, such as the theory of generalized functions.It is associated with such names as L. Schwartz, J.. Mikusiński and George. Temple.Within the framework of this theory was established clear and precise theoretical basis for such expressions as the Dirac delta function (it describes the region unified area, concentrated in an infinitesimal neighborhood of the point) and "step" Heaviside.Through this work Fourier series became useful for solving equations and problems, which involve intuitive concepts: point charge, point mass, magnetic dipoles, and the concentrated load on the beam.
Fourier method
Fourier series, in accordance with the principles of interference, begins with the decomposition of complex forms into simpler.For example, a change in the heat flux due to its passage through the various obstacles of insulating material of irregular shape, or a change in the earth's surface - an earthquake, a change in the orbit of a celestial body - the influence of the planets.Typically, these equations describing simple classical systems is elementary solved for each wave.Fourier showed that simple solutions can be summarized as for more complex tasks.In the language of mathematics, the Fourier series - a methodology for the submission of expression amount of harmonics - cosine and sine waves.Therefore, this analysis is also known as "harmonic analysis."
Fourier Series - an ideal method to the "computer age»
Prior to the creation of computer technology Fourier technique is the best weapon in the arsenal of scientists working with the wave nature of our world.Fourier series in complex form allows you to not only solve simple problems that lend themselves to direct application of Newton's laws of mechanics, but also the fundamental equations.Most of the discoveries of the nineteenth century Newtonian science became possible only due to the Fourier method.
Fourier series today
With the development of computers Fourier risen to a qualitatively new level.This technique is firmly entrenched in almost all fields of science and technology.As an example, a digital audio and video signal.Its implementation has been made possible only thanks to the theory developed by French mathematician in the early nineteenth century.Thus, the Fourier series in complex form has allowed to make a breakthrough in the study of outer space.In addition, it 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.In common cases, the number of such expressions can be endless.The greater the number considered in the calculation, the more accurate the final result is obtained.The most common use of simple trigonometric functions cosine and sine.In this case, the Fourier series is called trigonometric, and the decision of such expressions - harmonic decomposition.This method has an important role in mathematics.First of all, a trigonometric series provides a means to image and study the functions that it is the main unit of the theory.In addition, it allows us to solve a number of problems in mathematical physics.Finally, this theory has contributed to the development of mathematical analysis gave rise to a number of very important branches of mathematics (integral theory, the theory of periodic functions).In addition, the starting point for the development of the following theories: sets, functions of a real variable, functional analysis, and marked the beginning of harmonic analysis.
