In analysis, the Fourier series are a fundamental tool in the study of the periodic functions. It is starting from this concept that the branch of mathematics known under the name of developed analyzes harmonic.
The study of a periodic function by Fourier series comprises two aspects:
- the analyzes , which consists of the determination of the continuation of its coefficients of Fourier;
- the synthesis , which makes it possible to find, in a certain direction, the function using the continuation of its coefficients.
Beyond the problem of the decomposition, the theory of Fourier series establishes a correspondence between the periodic function and the coefficients of Fourier. So the analysis of Fourier can be regarded as a new way of describing the periodic functions. Operations such as derivation are written simply in terms of coefficients of Fourier. The construction of a periodic function solution of a functional equation can be reduced to construction corresponding coefficients of Fourier.
Fourier series were introduced by Joseph Fourier in 1822, but it was necessary one century so that the analysts give off the adapted tools of study: a theory of the integral fully satisfactory and the first concepts of the analyzes functional. They are still currently the subject of active research for themselves, and caused several new branches: analyzes harmonic, theory of the signal, Ondelette S, etc
Fourier series usually meet in the decomposition of periodic signals, the study of the electric currents, the cerebral waves, in the sound Synthèse, the Image processing, etc
A function F of a real variable is known as periodic of period T when it checks: for any reality X , . The frequency F is the reverse of the period: F=1/T .
The periodic functions of period T easiest to study are the sinusoidal functions whose frequency is a multiple of F
A linear Combinaison of these elementary sinusoidal functions bears the trigonometrical name of Polynôme and constitutes also a periodic function of period T . It can rewrite like linear combination of functions , the use of the complex numbers and the function Exponentielle making it possible to simplify the notations.
A trigonometrical polynomial P is thus written in the form:
Principle of Fourier series
For reasons of dimension, one cannot obtain all the periodic functions of period T like such a combination.
The idea subjacent with the introduction of Fourier series is to be able to obtain a function T - periodic, for example continues, like summons sinusoidal functions:
It is this time about a true infinite sum, i.e. of a limit of finished sum, which corresponds to the concept of nap of series.
Many calculations are translated in a very simple way on the coefficients of the trigonometrical polynomials, like the calculation of derivative. It is possible to generalize them on the level of the general coefficients of Fourier.
In a strict sense, the formula of decomposition is not correct in general. It is it, punctually, under good assumptions of regularity relating to F . Alternatively, one can give him direction while placing oneself in good functional spaces.
Fourier series constitute the oldest branch of the analyzes harmonic, but do not remain about it less one alive field, with the many open-ended questions. The study of their characteristics went hand in hand, during all the 19th century, with progress of the theory of the integration.
The first considerations on the trigonometrical series appear about 1400 in India, at Madhava, leader of the school of Kerala. In Occident, one finds them at the 17th century at James Gregory, at the beginning of the 18th century at Brook Taylor. It is the work of the latter, Methodus Incrementorum Directa and Inversa , appeared in 1715, which gives the kickoff to the systematic study of the vibrating cords and the propagation of the sound, major research topic during all the century.
A controversy bursts in the years 1750 between of Alembert, Euler and Daniel Bernoulli on the problem of the vibrating cords. D' Alembert determines the equation of wave and its analytical solutions. Bernoulli also obtains them, in the form of decomposition in trigonometrical series. The controversy relates to the need for reconciling these points of view with the questions of regularity of the solutions. According to J. - P. Kahane, it will have an important role in the genesis of Fourier series.
Bernoulli had introduced trigonometrical series into the problem of the vibrating cords to superimpose elementary solutions.
Joseph Fourier (1768 - 1830) introduces the equation of heat into a first report in 1807 which it supplements and present in 1811 for the Grand Prix of Mathematics. This first work, discussed in the field of the analysis, was not published. In 1822, Fourier exposes the series and the transformation of Fourier in his treaty analytical Théorie of heat . It states that a function can be broken up in the form of trigonometrical series, and that it is easy to prove the convergence of this one. He judges even any assumption of useless continuity.
In 1829, Dirichlet (1805 - 1859) gives a first statement correct of convergence limited to the continuous periodic functions per pieces having only one finished number of extrema. Dirichlet considered that the other cases were brought back there; the error will be corrected by Jordan in 1881.
In 1848, Wilbraham is the first to highlight the phenomenon of Gibbs while being interested in the behavior of Fourier series in the vicinity of the points of discontinuity.
Joint projection of Fourier series and the real analysis
The Memory on the trigonometrical series of Bernhard Riemann (1826 - 1866) , published in 1867, constitutes a decisive projection. The author raises a major hurdle by defining for the first time a theory of the satisfactory integration. He shows in particular that the coefficients of Fourier have a null limit ad infinitum, and a result of convergence known as the theorem of sommability of Riemann.
Georg Cantor (1845 - 1918) publishes a series of articles on the trigonometrical series between 1870 and 1872, where it shows its theorem of unicity. Cantor refines its results by seeking " whole of unicité" , for which its theorem remains checked. It is the origin of the introduction of the Set theory.
In 1873, Of Wood-Reymond (1831 - 1889) gives the first example of function continues periodic from which the Fourier series diverge in a point. The last quarter of the 19th century sees relatively few advanced in the field Fourier series or of the real analysis in general, whereas the Analyze complexes knows a fast progression.
In a note of 1900 and in an article of 1904, Fejér (1880 - 1959) shows its uniform theorem of convergence using the proceeded of summation of Cesàro (mean arithmetic of the sums partial of Fourier). Especially, it works out a new principle: systematic association between regularization by means of a “core” and Proceeded of summation for the Fourier series.
New tools of study
Henri Lebesgue (1875 - 1941) gives to the theory Fourier series his final framework by introducing a news theory of integration. In a series of publications which are spread out 1902 with 1910, it extends the theorems of its predecessors, in particular the theorem of Riemann on the limit of Fourier series. It also proves several new theorems of convergence. The majority of its results appear in its Leçons on the trigonometrical series published in 1906.
In 1907, Pierre Fatou (1878 - 1929) shows the equality of Parseval within the general framework of the functions of summable square. The same year, Frigyes Riesz (1880 - 1926) and Ernst Fischer (1875 - 1954) , independently, proves the reciprocal one. These results take part in the birth of a new field, the analyzes functional.
Henceforth, the questions of convergence in functional spaces are considered through the study of the properties of the continuations of cores and the associated operators. Most of the results passes by questions of estimate of standards called " constants of Lebesgue" , which becomes a systematic object of study.
In parallel, the problem of the simple convergence of Fourier series gives place to several dramatic turns of events with the publication of results which knew a great repercussion and surprised the contemporaries. In 1926, Andreï Kolmogorov (1903 - 1987) builds an integrable example of function from which the Fourier series diverge everywhere. In 1966, Lennart Carleson (1928) establishes on the contrary that the Fourier series of a summable function of square converge almost everywhere towards this function. Other results (Kahane and Katznelson 1966, Hunt 1967) come to supplement the study. Research goes then on the convergence of Fourier series to several dimensions, still imperfectly known.
Coefficients of Fourier
The definition of the coefficients of Fourier carries on the periodic functions integrable within the meaning of Lebesgue over one period. For a periodic function, being of class implies the integrability. This includes/understands in particular the continuous, or continuous functions by pieces, periodicals. Here the notations of the first paragraph are taken again.
The coefficients of Fourier (complex) of F (for ) are given by:
If n>0, one calls Harmonique row N of the function F the sinusoidal function of frequency N F obtained by taking account of the coefficients of Fourier of index N and - N, given by:
The Fourier series of F are the series of functions obtained by summoning the successive harmonics, in other words the Série of functions:
If the function F is with actual values, it can be interesting to handle real coefficients, in particular in the case of even or odd functions. One thus defines the real coefficients of Fourier of F :
- For N >0, ;
- For N >0, .
There still, the periodicity authorizes to change the interval of integration.
The Harmonique of row N is rewritten then like the function:
Following convention can also be selected for a0 : , what is not interpreted then any more like a median value, but is the double. This last convention harmonizes the definitions of the coefficients which then begin all with .
The systems of coefficients ( has N , B N ), for positive N , and C N , for relative N whole are linearly dependant by the following relations:
- , ;
- For , and .
The parity of a function is translated on the coefficients of Fourier:
- a function F is even, if and only if c-n (F) =cn (F) for all N . In the case of a real function F , this property becomes bn (F) = 0 for all N .
- a function F is odd, if and only if c-n (F) =-cn (F) for all N . In the case real that gives an (F) = 0 for all N .
Equality of Parseval
See also: equality of Parseval
For a function T - periodic continuous per pieces, or more generally of integrable square over one period, the equality of Parseval affirms the convergence of the following series and the identity:
The equality of Parseval implies in particular that the coefficients of Fourier of F tend (sufficient quickly) towards 0 in the infinite one. According to the assumptions of regularity on F , the speed of convergence can be specified (see below).
Effect of derivation on the coefficients
For a continuous function and per pieces, one establishes, by Intégration by parts:
Coefficients and regularity of the function
The coefficients of Fourier characterize the function: two functions having the same coefficients of Fourier are equal almost everywhere . In particular, in the continuous case per pieces, they coincide in all the points except a finished number.
A certain number of results connect regularity of the function and behavior ad infinitum coefficients of Fourier.
- the Théorème of Riemann-Lebesgue watch that the coefficients of Fourier of a function F integrable over one period tend towards 0 when N tends towards the infinite one.
- the identity of Parseval admits reciprocal: a function is of square summable over one period if and only if the series of the squares of the modules of the coefficients of Fourier converges. It is the Théorème of Riesz-Fischer.
- There exists little of similar characterizations for other functional spaces. One can affirm however that a periodic function is if and only if its coefficients of Fourier are with fast decrease,