Biography of Joseph Fourier

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Jean-Baptiste Joseph Fourier was born on the 21st of March 1768 in the ancient town of Auxerre, Burgundy, Kingdom of France (now in Yonne, France). This small town has many cultural, religious, political and educational traditions. It was one of the foremost centers of teaching and learning in France in the ninth and tenth centuries. It is located on the bank of the river Yonne with many magnificent tall building including the famous ancient Clock Tower, the even more famous Abbey St. Germain founded by St. Germain himself in the fifth century A.D., and the Gothic Cathedral St. Etienne. It became more famous as the birthplace of Joseph Fourier, one of the most brilliant mathematical scientists France has ever produced.

Joseph’s father was a master tailor of Auxerre. After the death of his first wife, with whom he had three children, he remarried and Joseph was the ninth of the twelve children of this second marriage. Joseph's mother was a housewife. When he was nine- or ten-years old, both his parents died. He first entered the Pallais elementary school to receive lessons in Latin and French. He was a gifted student in both languages, but showed extraordinary talents for mathematics and mechanics. At the age of 13, he had completed an extensive study of six volumes of Etienne Bézzout’s (1730--1783) Cours de mathématique. In 1783, he received the first prize for his study of C. Bossut’s (1730--1814) Mécanique. At a very young age, Joseph presented a research paper on algebraic equations at the Académie Royale des Sciences. In 1787, Fourier decided to train for the priesthood and entered the Benedictine abbey of Saint-Benoit-sur-Loire. Later, however, he gave up his idea of studying theology. He was then admitted to the progressive École Royale Militaire to study science and mathematics. In 1790, he became a mathematics teacher at the Benedictine College, École Royale Militaire of Auxerre, where he had been a student.

The French Revolution was a watershed event in modern European history that began in 1789 and ended in the late 1790s with the ascent of Napoleon Bonaparte. In 1789, Joseph returned to Auxerre and received a teaching position in his old school. Fourier took an active role in the French Revolution, as he was keenly attracted to its egalitarian ideals. He joined his local Revolutionary Committee, but soon regretted it. During his time in the Committee, he made the mistake of confronting a rival sect in defense of his own faction while on a trip to Orléans, consequences of which would later threaten his life. Certainly, Fourier was unhappy about the Terror that resulted from the French Revolution and he attempted to resign from the committee. However this proved impossible and Fourier was now firmly entangled with the Revolution and unable to withdraw. After Fourier's return to Auxerre, he continued to work on the Revolutionary Committee and continued to teach at the College. In July 1794 he was arrested by the order of the Committee for Public Safety for his possible revolutionary and terrorist activities in the Revolution in Auxerre during 1793--1794, the charges relating to the Orléans incident, and he was imprisoned. Fourier feared that he would go to the guillotine but, after Maximilien Robespierre himself went to the guillotine, political changes resulted in Fourier being freed.

Later in 1794, now free, Joseph was nominated to study at the École Normale in Paris, which opened in 1795 as a model school for training teachers in France. He joined this teacher-training school and got a rare opportunity to meet some of the foremost French mathematicians of that time including J.L. Lagrange (1736--1813), P.S. Laplace (1749--1827) and Gaspard Monge (1746--1818). In September 1795, Fourier was appointed as an assistant lecturer at the École Polytechnique to support the teaching of Lagrange and Monge. In a short period of two years, he succeeded Lagrange to occupy the Chair of Analysis and Mechanics at the Polytechnique in 1797. In spite of his cordial relationship with Lagrange, Laplace and Monge, and his exceptional teaching record, Fourier’s research record at the time was not that outstanding by any standard.

In 1798, Fourier was selected to join Napoleon Bonaparte’s (1769--1821) Egyptian expedition as scientific adviser to liberate unhappy Egyptian people from deep social and political problems and to provide them with all the benefits of European civilization (NB Fourier himself was opposed to the abusive treatment inflicted by the colonial rulers, and during his stay in Egypt, worked almost solely within the field or pure mathematics). He remained in this position until the downfall of Napoleon in 1815. During his occupation of Egypt, Napoleon established the Institute d’Egypt in Cairo with Monge as its President and Fourier as Permanent Secretary. Fourier continued his position during the entire occupation of Egypt and was fully responsible for collating all literary and scientific discoveries made during that time. In addition, he was very active in administering the archaeological research of the Cairo Institute. This position led Fourier to publish the Description de l’Egypte (Description of Egypt) with an elegant preface, partly edited by Napoleon himself.

In 1799, Napoleon returned to Paris to hold the absolute power of France and subsequently crowned himself Emperor of the French. In 1801, Fourier also came back to Paris to resume his position as Professor of Analysis at the École Polytechnique. After his return from Egypt in 1802, Fourier was appointed Prefect of Isére in Grenoble with many major and varied administrative duties and responsibilities. As Prefect from 1802 to 1815, Fourier was totally successful in planning, management and the draining operation of the largest swamps of Bourgoin, as the swamps had been fully responsible for annual epidemics of dangerous fever causing deaths of a large number of the surrounding inhabitants at a young age. After completion of the draining of the swamps in 1812, the annual epidemics of fever stopped leading to a marked improvement in health problems of the inhabitants. This was one of the greatest public and social service accomplishments of Fourier. His other major accomplishment as Prefect was the planning of a very long, spectacular highway from Grenoble to Turin, of which only the French Section was built. Despite his unfriendly relationship with Napoleon, Napoleon awarded Fourier the title of Baron in recognition of his major accomplishments as Prefect.

During 1787--1810, a series of personal letters between Fourier and C.L. Bonard (1765--1819), a mathematics teacher at Auxerre provided the unique opportunity to establish a real and permanent friendship between them. These letters were full of their many personal and family experiences and reminiscences of the old days, especially during the revolution in Auxerre. As a family friend and true admirer of Fourier, Bonard was always very proud of Fourier’s mathematical and other achievements. Unfortunately, Bonard died in 1819 just three years before the publication of Fourier’s masterpiece work on the Theory of Heat.

This masterpiece, which included his work on the Fourier series, was actually finished in 1807 as part of his publication 'On the Propagation of Heat in Solid Bodies'. It was not published until 1822, however, for his first reading of it (to the Paris Institute) in 1807 was met with very mixed and critical reception. Both Lagrange and Laplace objected to the notion of what we now call Fourier series: the expansions of functions as trigonometrical series. Along with another scientist, Jean-Baptiste Biot (1774--1862), Lagrange and Laplace also objected to Fourier’s derivation of the equations of transfer of heat. (Biot had written an earlier paper on the topic in 1804, although that paper proved incorrect.) Nonetheless, when the Paris Institute held a competition on the topic of how heat propagates in solid bodies in 1811, Fourier submitted his memoir for consideration. He won the prize, in part because only one other entry was received.

Because of the controversy, Fourier’s memoir was not published until 1822, after his election to the Académie des Sciences in 1817, and the same year he became the Académie’s secretary. His work did contain flaws, but it also provided the basis for later work on trigonometric series and the theory of functions of a real variable, most notably the Fourier transform, an operation that turns one function of a real variable into another. It is widely used in digital signal processing, as well as in the physical study of wave motion and optics.

In 1827, like Jean D’Alembert (1717–1783) and Laplace before him, Fourier was elected to the literary Académie Francaise. In the same year, after the death of Laplace, Fourier was also elected President of the Conseil de perfectionnement of the École Polytechnique. Several foreign scientific organizations including the Royal Society of London and the Royal Swedish Academy of Sciences elected Fourier as honorary member for his outstanding contributions to mathematical sciences. Fourier continued to publish papers on mathematics until his death in 1830, when he tripped and fell down the stairs at home.

Throughout his life, Fourier’s general health was not very good. Towards the end of his life, he began to display some unusual symptoms which were thought to have possibly been contracted during his stay in Egypt from the extremely hot climate. In early May of 1830, Fourier suffered from a heart attack and his condition gradually deteriorated until he died on May 16 at the age of 62. He was buried at the cemetery of Père Lachaise in Paris. His gravestone was decorated with an Egyptian motif to recognize his outstanding service as Permanent Secretary of the Cairo Institute and his collation of the landmark Description de l'Égypte.

 

During his stay in Grenoble, Fourier found the time and intellectual energy to discover the theory of heat conduction based on his new mathematical ideas, methods, experiments and observations. He completed his classic publication on the conduction of heat entitled ‘On the propagation of heat in solid bodies’, and submitted it to the Academy of Sciences of Paris in 1807 for a research prize. His article was judged by a great scientific committee consisting of Lagrange, Laplace, Monge, A.M. Legendre (1752--1833) and S.L. Lacroix (1765--1843). The committee was generally impressed by the importance and novelty of Fourier’s work, but it was split on the overall quality and rigor of the mathematical treatment involving the use of trigonometric (Fourier) series expansions of functions without adequate justification. The controversy created by his work on trigonometric expansions of functions was due to his demonstration of the paradoxical property of equality in a finite interval between algebraic results of a totally different form. Another major criticism of his publication was about his derivation of the heat equation in a continuous solid was based on inadequate physical principles. Unfortunately, his memoir was rejected by the committee for the prize. Although some of his contemporary mathematicians vociferously objected to his revolutionary work, the Academy was generally impressed by the subject matter of Fourier’s memoir and encouraged him to resubmit it for a Grand Prize to be awarded in 1812. At the same time, S.D. Poisson (1781--1840) and J.B. Biot (177--1862), his rivals, expressed serious concerns with major criticisms of Fourier’s work on his mathematical and physical treatment of the theory of heat in solids. Surprisingly, Biot praised Poisson’s less original work on the same subject. Poisson’s harsh criticism is quoted below in full:

As the partial differential equation to which it corresponds is linear and has constant coefficients, one can also satisfy it by an integral composed of an infinity of exponentials of sines and cosines containing an infinite number of arbitrary constants: this integral is contained in the preceding one; but it would be difficult to decide a priori if it has the same degree of generality and if it can replace it identically, something which necessarily throws doubt and obscurity on all solutions deduced from this second form of the integral. M. Fourier, who did not go beyond a solution of this kind, remarks himself that it is similar to that which Daniel Bernouilli gave to the problem of vibrating strings; but it is well known that Euler, d’Alembert, and Lagrange, who occupied themselves at the same time with the same problem, and who differed among themselves on various points, were at one nevertheless in regarding Bernouilli’s solutions as incomplete and less general than that containing arbitrary functions. This is not true of the formulas of M. Fourier: I am sure that all the results he obtains are correct; but against his analysis can be advanced the same objections as those advanced against that of Bernouilli and repeated in other similar cases.

In general it seems to me that whenever an unknown quantity depends on a partial differential equation, and when its values should reduce in fact to a sum of particular integrals, the only way of disposing of all doubts and retaining for the mathematical certainty result is not to suppose in advance such a form for the unknown quantity, but to deduce it, on the contrary, from the general integral by a succession of direct and rigorous transformations. This is what I have attempted to do in this memoir . . .

. . . I leave it to mathematicians to judge if I have attained the end that I have set myself.

Fourier had real talent, and extraordinary courage to defend his work in an absolutely unprecedented manner. His devastating response to Poisson and Biot’s unfounded criticisms of his solution was contained in his unpublished Historical Pré́cis, and in his letters to Laplace around 1808--1809 with a note on certain analytic expressions in connection with the equations of the theory of heat. Thereafter, all criticisms seemed to have subsided once, and for all, barely having tarnished his permanent reputation.

Fourier continued his research and resubmitted his revised article in 1811 with some addition of new material on the cooling of infinite solids, radiant and terrestrial heat, comparison of his theory with experimental observations and equations of the movement of heat in fluids. In spite of some disagreements, Fourier was awarded the Grand Prize for his work in 1812, but his work was not recommended for publication in the Mé́moires of the Academy. In 1817, he was elected member of the Acadé́́mie des Sciences by an overwhelming majority. He continued to revise his work for publication. In 1822, J.B. Delambre (1749--1822), who was Permanent Secretary of the mathematics and physics division of the Acadé́mie des Sciences died, and Fourier was elected as his replacement. Fourier remained in this position until his death in 1830. Shortly after he became the Secretary, the Acadé́mie published his masterpiece work entitled La Thé́́orie Analytique de la Chaleur (The Analytical Theory of Heat) based on Isaac Newton’s (1642--1727) law of cooling, that the flow of heat between two adjacent molecules is proportional to extremely small temperature difference. In his preface of this great book, Fourier described his own assessment of the work as follows:

First causes are not known to us, but they are subjected to simple and constant laws that can be studied by observation and whose study is the goal of Natural Philosophy . . . Heat, like gravity penetrates every substance of the universe; its rays occupy all regions of space. The object of our work is to set forth the mathematical laws that this element obeys . . . But whatever the extent of the mechanical theories, they do not apply at all to the effects of heat. They constitute a special order of phenomena that cannot be explained by principles of movement and of equilibrium . . . The differential equations of the propagation of heat express the most general conditions, and reduce physical questions to problems in pure analysis that is the proper object of the theory.

His analytical theory of heat was highly mathematically based on a general equation of propagation of heat subject to appropriate initial and boundary conditions of different kinds. Fourier’s fundamental heat equation for the temperature distribution, u = u(x, t) is given by

where x = (x,y,z), \( \kappa = \frac{k}{c\,\rho} \) is the diffusivity constant, k is the heat conductivity, c is the specific heat, ρ is the density, and \( \nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} + \frac{\partial^2}{\partial z^2} \) is the three-dimensional Laplacian. Thus, the heat equation is of the first order in time derivative and of the second order in space derivatives. This was a striking contrast with almost all dynamic equations, including the wave equation in strings or air that are of second order in both time and space derivatives.

One of the French scientific philosophers and close colleagues of Fourier, Auguste Compte (1798--1857) attended a course of lecture of Fourier on the role of the positivist philosophy of science. In his published book entitled Cours de Philosophie positive dedicated to Joseph Fourier, Compte expressed his eloquent and complimentary views on Fourier’s celebrated theory of heat as follows:

In fact, in this work, of which the philosophical characteristic is so eminently positive the most important and precise laws of thermal phenomena are discovered without the author having once inquired about the intimate nature of heat.

Subsequently, Henri Poincaré́ (1854--1912), an extremely versatile and creative mathematician and mathematical physicist, made a delightful comment on Fourier’s fundamental work as follows:

Fourier’s theory of heat is one of the first examples of the application of analysis to physics. Starting from simple hypotheses, which are nothing but generalized facts, Fourier deduced from them a series of consequences which together make up a complete and coherent theory. The results which he obtained are certainly interesting in themselves, but what is still more interesting is the method which he used to arrive at them and which will always be a model for all those who wish to cultivate any branch of mathematical physics.

Even with lack of rigor in Fourier’s classic work on the analytical theory of heat, a eminent British physicist, William Thomson, 1st Baron Kelvin (1824--1907) regarded the work as Fourier’s permanent legacy and called it ‘a great mathematical poem.’

 

This section includes some of Fourier’s major contributions to other areas of mathematics and physics and their foremost impact on mathematical physics, mathematical economics, probability, statistics, pure and applied mathematics. During 1817--1818, he made notable contributions to the theory of Fourier transforms and their applications to partial differential equations including the heat and the wave equations. His 1822 monumental treatise on the Analytical Theory of Heat provided the modern mathematical theory of conduction of heat, Fourier series and Fourier integrals with many new examples of applications. In his treatise, he formulated a remarkable theorem which is now universally known as the Fourier Integral Theorem that states that an arbitrary function f(x) has the Fourier double integral representation in the form

This representation shows that f(x) has been decomposed into an infinite number of waves of different wavenumbers k (or wavelengths λ = 2π/k), and amplitude \( \frac{1}{2\pi} \int_{-\infty}^{\infty} {\text d}\xi\, f(\xi )\, e^{-{\bf j}kx} , \) whereas a given function can be represented by a Fourier series in terms of an infinite but discrete set of wave components.

This theorem has been expressed in several slightly different forms to better adapt it for many particular applications. It has been recognized, almost from the start, however, that the form which best combines mathematical simplicity and complete generality makes use of the exponential oscillating function exp(jkx). Indeed, the Fourier integral formula is regarded as one of the most fundamental results of modern mathematical analysis, and it has widespread physical and engineering applications. The generality, applicability and importance of the theorem is eloquently expressed by Lord Kelvin and Scottish mathematician Peter Guthrie Tait (1831--1901) as follows: ‘. . . Fourier’s Theorem, which is not only one of the most beautiful results of modern analysis, but may be said to furnish an indispensable instrument in the treatment of nearly every recondite question in modern physics. To mention only sonorous vibrations, the propagation of electric signals along a telegraph wire and the conduction of heat by the earth’s crust, as subjects in their generality intractable without it, is to give but a feeble idea of its importance’. This integral formula is usually used to define the classical Fourier transform of a function and the inverse Fourier transform. No doubt, the scientific achievements of Joseph Fourier have not only provided the fundamental basis for the study of heat equation, Fourier series and Fourier integrals, but also for the modern developments of the theory and applications of the partial differential equations.

Although Fourier is most celebrated for his work on the conduction of heat, Fourier series and Fourier transforms, the new mathematics he created has proved to be very useful in a wide variety of ways. His superb mastery of analytical treatment and physical intuition played a fundamental role in his success. In mathematical physics, Fourier’s contributions have proved to be much more useful than those of his renowned contemporaries including Lagrange, Laplace, Poisson, Biot and Augustin Cauchy (1789–1857). Both Cauchy and Poisson successfully applied Fourier’s universal mathematical analysis to the subject of linear water waves and to other areas. It has also been a remarkable fact that Fourier analysis eventually resolved the long-standing controversy over the mathematical treatment of the famous vibrating string problem which was vigorously investigated by several eminent mathematical scientists including Daniel Bernoulli (1700–1782), Leonhard Euler (1707–1783), D’Alembert and Lagrange. Fourier’s work was also remarkably influential for the comprehensive development of James Clerk Maxwell’s (1831–1879) electromagnetic theory and kinetic theory of gases. In 1864, Maxwell first formulated the basic equations for all electromagnetic phenomena, known as the Maxwell equations. Maxwell expressed his full admiration for Fourier’s great work on the theory of heat.

Among his other research contributions to physical sciences, Fourier first suggested that the earth’s atmosphere helps trap the heat of sunlight. This special behavior of the atmosphere is known as the greenhouse effect because it resembles the action of the green roof of a greenhouse. In his series of articles published during 1824 and 1827, he investigated various possible sources of the additional observed heat, and suggested that interstellar radiation may be responsible for a large amount of warmth. Apart from this work, Fourier published many papers on terrestrial heat and radiant heat which led to his mathematical derivation of the sine law for emission of radiation at the surface of heated bodies based on John Leslie’s (1766--1832) experimentally determined the sine law. He also studied the application of the principle of virtual velocities to fluid flows, oscillations of system of bodies about the state of equilibrium, and the subject of wave motions in elastic lamina. He also took up a few aspects of Laplace’s work on probability and statistics dealing with the significance of the normal distribution and the central limit theorem. In particular, his interest in the density function \( f(x) = \frac{1}{\sqrt{2\pi}} \,\exp \left( - x^2 /2 \right) \) of the 2 standard normal (or Gaussian) distribution occurs in a wide variety of mathematical contexts, from probability theory to Fourier analysis to quantum mechanics. This density function has some unique properties that are not shared by any other special functions in mathematics and mathematical physics.

 

Historically, it was Fourier who first not only represented an arbitrary function in trigonometric series of cosine and sine functions, but also raised the fundamental question of convergence of the series. Once he made the following bold statements: ‘Thus, there is no function . . . which cannot be expressed by a trigonometric series . . . [or] definite integral’. ‘Regarding the researches of d’Alembert and Euler could one not add that if they knew this expansion they made but a very imperfect use of it. They were both persuaded that an arbitrary and discontinuous function could never be resolved in series of this kind, and it does not seem that anyone had developed a constant in cosines of multiple arcs, the first problem which I had to solve in the theory of heat’. The first major result about the convergence of a series of functions was Peter Gustav Lejeune Dirichlet’s (1805--1859) theorem of convergence of Fourier series of piecewise continuous functions. In 1829, he was one of the first mathematicians who proved the first theorem giving sufficient (and very general) conditions for the Fourier series of function f(x) to converge pointwise to f(x) after 18 years of Fourier’s discovery. The proof given by Dirichlet was a kind of refinement of that sketched by Fourier in the final section of his treatise on the theory of heat. Dirichlet’s method of proof was to calculate the limit of the nth partial sum of a Fourier series, s_n(x) as n → ∞. He proved that for any given value of x, the sum of the Fourier series is f(x) if f(x) is continuous at that point x, and is \( \frac{1}{2} \left[ f(x+0) +f(x-0) \right] \) if f(x) is discontinuous at that point. It was Dirichlet who closely followed Fourier’s work and then in 1829 established for the first time a rigorous mathematical theory of Fourier series. In 1835, he also established the Poisson summation formula for the Fourier analysis, and then modernized the concept of function that served as the fundamental basis for the nineteenth century investigations of mathematical analysis. In connection with the study of convergence of Fourier series, Dirichlet commended Fourier’s great work: ‘The sine and cosine series, by which one can represent an arbitrary function in a given interval, enjoy among other remarkable properties that of being convergent. This property did not escape the great geometer (Fourier) who began, through the introduction of representation of functions just mentioned, a new career for the applications of analysis; it was stated in the article that contains his first research on heat. But no one so far, to my knowledge, gave a general proof of it’. Inspired by Dirichlet’s work on Fourier series, Bernard Riemann (1826--1866) first formulated the concept of Riemann integral in his memoir on trigonometric series in 1854 that served the fundamental basis for his profound work on the convergence of Fourier series.

According to his fundamental theorem, for a bounded integrable function f(x), the convergence of its Fourier series at a point x in [-π, π] depends only on the behavior of f(x) in an arbitrarily small neighborhood of that point x. It was Riemann who first observed this behavior and referred it as the localization principle for the convergence of Fourier series. Both Dirichlet and Riemann recognized that more than continuity is required to ensure the convergence of Fourier series, namely the requirement of bounded variation of the function f(x) in a neighborhood of x. In 1854, Riemann praised Fourier’s work and said: ‘Nearly fifty years had passed without any progress on the question of analytic representation of an arbitrary function, when an assertion of Fourier threw new light on the subject. Thus a new era began for the development of this part of Mathematics and this was heralded in a stunning way by major developments in mathematical Physics’. Further investigations, continuing to the present, led to extensions of this result and revealed that the general question of convergence of a Fourier series is a very difficult and deep subject in mathematical sciences. However, the problem of finding necessary and sufficient conditions on f(x) so that its Fourier series converges to f(x) remained unsolved for a period of century. The nature of the convergence of Fourier series also received further attention after the introduction of the notion of uniform convergence by several leading mathematical scientists including George Gabriel Stokes (1819--1903), Philipp L. Seidel (1821--1896) and Karl Weierstrass (1815--1897). Subsequently, based on measure theory Henri Lebesgue’s (1875--1941) theory of integral and convergence in L²-norm were put Fourier analysis on the modern mathematical map. His discovery of these fundamental ideas and methods undoubtedly stimulated further development of Fourier series and integrals. Despite various attempts, it was an open question and unsolved problem for a period of a century whether a Fourier series of a continuous function converges at a point. In 1915, a Russian mathematician Nikolai Luzin (1883--1950) conjectured that if f ∈ L²[0, 2π], then its Fourier series converges almost everywhere. In 1966, Lennart Carleson (born in 1928) provided an affirmative answer of Luzin’s conjecture with the deepest theorem which states that the Fourier series of any square integrable function f(x) in [0, 2π] (i.e. f ∈ L²[0, 2π]) converges almost everywhere to f(x). Shortly, after Carleson’s great work, Hunt proved that almost everywhere convergence of Fourier series of functions in L^p[0, 2π] for 1 < p < ∞. One year later, Sjolin gave further refinement of Hunt’s theorem to include spaces of functions that are smaller than L¹, but larger than L^p for every p > 1. Subsequently, Hunt and Taibleson proved the sharpest result on almost everywhere convergence of Fourier series on the ring of integers of a local field. However, another great mathematician of the twentieth century, A.N. Kolmogorov’s (1903--1987) 1926 classical counter example shows that the result fails when p = 1, that is, the Fourier series of L¹ functions diverges everywhere. This resolved an important question of convergence of Fourier series which became very useful in mathematical analysis and in many other areas of mathematical sciences.

 

Fourier had early strong interests in pure mathematics. In fact, Bezout’s book on Théorie gé́nérale des equations algébbraiques (1779) stimulated Fourier’s active research interest in classical algebra. At the age of 21 in 1789, he presented a paper on algebraic equations at the Académie Royale des Sciences. He gave a new proof of the famous rule of René Descartes (1596--1650) which describes how to determine the number of positive and negative roots of algebraic polynomial equations. His lifelong interest in the theory of equations was evident from his incomplete book manuscript on determinate equations which was edited by his close friend and a famous applied mathematician, Claude-Louis Navier (1785--1836) and then published it posthumously with the title Fourier’s Analyse des équations détterminées in 1831. Fourier’s other ideas on algebra eventually led to his student Charles Sturm’s (1803--1855) basic theorem. The solution of Fourier’s general equation of heat conduction in two or three dimensions required precise boundary conditions. Fourier’s first mathematical treatment and then by his students Charles Sturm and Joseph Liouville (1809–1882) opened up the whole new field of eigenvalues and eigenfunctions of ordinary and partial differential equations of such enormous importance for modern applied mathematics and mathematical physics. Apparently, Fourier himself realized the extraordinary nature and true significance of his discovery of the equation of heat conduction and its methods of solution by Fourier series and Fourier transforms. This work displayed his unique physical understanding and mathematical ingenuity. Other modern developments in pure mathematics which can be traced back to Fourier’s analytical theory of heat included the theory of orthogonal functions, Fourier’s expansion of an arbitrary function in terms of eigenfunctions, the ideas of different types of convergence, definite integral as the limit of a sum and theory of infinite determinants. Thus, Fourier’s most important, and, indeed, revolutionary contribution to pure mathematics was the representation of functions by trigonometric (Fourier) series, its numerous mathematical properties including the theory of convergence, term-by-term integration and differentiation, and then, the determination of the sum of numerical infinite series which is often impossible by other methods of summation of infinite series. There are examples which showed that the constructed Fourier series whose values on a given interval may not, in general, be the same as those of the function.

Albert Einstein (1879--1955) well-described the mutual relationship of pure mathematics, applied mathematics and physical laws as: ‘pure mathematics enables us to discover the concepts and laws connecting them, which gives us the key to the understanding of the phenomena of nature’.

Fourier went even further than this by insisting on the way in which situations observed in nature stimulate advanced study and research in pure mathematics: ‘The profound study of nature is the most fertile source of mathematical discoveries. Not only does this study, by offering a definite goal to research, have the advantage of excluding vague questions and futile calculations, but it is also a sure means of moulding analysis itself, and discerning those elements in it which it is still essential to know and which science ought to conserve. These fundamental elements are those which recur in all natural phenomena’. He was right to make such a profound statement: some of the most abstract pure mathematics involved in the theory of Fourier series and Fourier transforms have been used successfully to solve many applied physical problems including the vibration of elastic strings, and to develop the analytical theory of flow of heat in solid bodies of different geometric configurations. During the subsequent development of Fourier series, the English mathematician Henry Wilbraham (1825--1883) gave a remarkable surprise by his discovery in 1848 what is now known as the Gibbs phenomenon (called by Maxime Bôcher after Joseph Willard Gibbs who discovered it in 1899) which states that near a jump discontinuity of a function, its Fourier series overshoots (or undershoots) it by approximately 9% of the jump.

Admittedly, the idea of convolution of two or more functions arose naturally first in the study of Fourier series and Fourier transforms. It also played a fundamental role in Fourier analysis, number theory, probability theory, mathematical statistics, harmonic analysis and served more generally in mathematical analysis of functions in other settings. Georg Cantor (1845--1918) revolutionized mathematics in the second half of the nineteenth century by creating the foundation of the set theory which was motivated in part by questions of convergence and uniqueness of trigonometric and Fourier series expansions. In the 1920s, Frigyes Riesz (1880--1956) established the norm convergence for Fourier series, and almost simultaneously, Ernst Fisher (1875--1959) unequivocally proved the mean-square convergence theorem for Fourier series. All these developments led to the theory of singular integral operators at the center of harmonic analysis. Fourier analysis, in general and Fourier transforms, in particular, have been very useful for the solution of linear partial differential equations with initial and boundary conditions of different kinds.

 

It was shown by Albert Einstein that the Fourier heat equation is closely associated with the diffusion process of Brownian motion. In 1828, the Scottish botanist Robert Brown (1773–1858) observed the continuous random movement of small particles of pollen suspended in water under the microscope. This never-ending very irregular and wiggly motion caused by collisions of molecules is known as Brownian motion. It was also found in colloidal particles in chemistry and in many other physical and biological phenomena. Brown pointed out that this irregular motion appeared to obey no known mathematical or physical laws. During the nineteenth century, several attempts had been made to understand the origin of the Brownian motion in microscopic living animals and colloidal particles in chemistry. It has been proposed that the Brownian motion of particles was caused by the electrostatic forces among them, and the constant changes in direction in motion took place due to the influences on particles from molecules of the surrounding medium. For sufficiently small light particles, numerous collisions could have microscopic impact on their random displacements. From a thermodynamic point of view, the thermal excitation of molecules is increased due to increase in temperature of the system, and hence, the physical explanation of Brownian motion is consistent with experimental observations. Fourier suggested that heat propagates almost like a collection of infinitesimal Brownian particles, and hence, his diffusion equation is closely related to Brownian motion.

In one of his three famous 1905 papers, Einstein developed a major physical theory of the random motion of the individual particles in Brownian motion. He described the collective motion of Brownian particles of density ρ(x, t) at position x and time t which satisfies the Fourier diffusion equation, \( \rho_t = \kappa\,\rho_{xx} \) with the solution

where κ is called the diffusion constant which measures how rapidly the distribution spreads out with time. In order to verify Einstein’s theoretical molecular explanation of Brownian motion, Jean Perrin (1870--1942) performed experiments and observed that functions describing the paths of Brownian motion are nowhere differentiable, even though they are continuous.

Using statistical methods together with heat equation, Einstein showed that if Brownian motion of particles starts at the origin, then after a fixed time t, its position is randomly distributed according to the three-dimensional Gaussian distribution with mean zero and variance κt. In other words, the mathematical description of the Brownian motion of particles in terms of a random process B_t with transitional probability densities is given by the diffusion (or Gaussian) kernel G_t (x, m) in n dimensions in the form

Since diffusion process results from the Brownian motion of particles, the exact solution u(x, t) of the Fourier diffusion equation in ℝⁿ with the initial data u(x,0) = u₀(x) is equal to the expected value of \( u_0 \left( \sqrt{2}\,B_t^x \right) , \) that is, where \( B_t^x \) is the n-dimensional Brownian motion starting at x. It is further noted that a trajectory of Brownian motion is almost surely non-differentiable. This explains the irregular randomness of the Brownian motion. Thus, the expected value result \( E \left[ u_0 \left( \sqrt{2}\,B_t^x \right)\right] \) describes the relation between the solution of heat equation and probability theory and stochastic processes. Subsequently, this relationship led to the celebrated Feynman--Kac formula: which relates the Brownian motion of particles to the solution of the n-dimensional heat equation with a given initial data, u₀ (x), where ∇²_n is the n-dimensional Laplacian. Subsequently, the idea of stochastic process or stochastic convergence of probability distributions came into existence when interpreted as modes of convergence of their Fourier transforms. Norbert Wiener (1894--1964) first developed the mathematical theory of probability measure and integration in function space to construct rigorous mathematical models for Brownian motion which was till then understood only heuristically by Einstein and others. He also constructed a probability measure in the space of all continuous functions on C[0, ∞), universally known as the Wiener measure. In fact, he discovered the systematic mathematical description of Brownian motion known as the Wiener process (a continuous-time stochastic process) in 1923 based on the theory of Fourier series.

Subsequently, several great mathematical scientists including Andrey Kolmogorov (1903--1987), Paul Levy (1886--1971), and Kiyoshi Itô (1915--2008) made some major contributions to stochastic analysis for a further understanding of Brownian motion and diffusive processes. Based on Fourier transforms, Levy formulated a new and simpler approach to the general theory of stable probability distributions. From a thermodynamic point of view, Brownian motion is irreversible in the sense that, for a large number of diffusing particles, there exists a smoothing of the probability distributions describing the state of particles as time increases. On the other hand, Itô discovered a powerful method of stochastic calculus and stochastic differential equations for an explicit description of the Markov diffusion process.

 

We next give some general idea of the broader impact Fourier analysis has had on some selected areas of mathematics, mathematical physics and mathematical economics. Using Parseval’s formula for Fourier series, Adolf Hurwitz (1859--1919) gave a very elegant solution of the famous isoperimetric problem in geometry. He proved that the area A of the region enclosed by a simple closed curve Γ in ℝ² of length ℓ satisfies the so-called isoperimetric inequality

where equality holds if Γ is a circle. Hurwitz made some comments praising the use of Fourier series in geometry: ‘Fourier series and analogous expansions intervene very naturally in the general theory of curves and surfaces. In effect, this theory, conceived from the point of view of analysis, deals obviously with the study of arbitrary functions. I was thus led to use Fourier series in several questions of geometry, and I have obtained in this direction a number of results which will be presented in this work. One notes that my considerations form only a beginning of a principal series of researches, which would without doubt give many new results’.

Several great mathematicians including Bernhard Riemann (1826--1866), Karl Weierstrass (1815--1897) and G.H. Hardy (1877--1947) investigated the famous problem of everywhere continuous but nowhere differentiable function in real analysis. In 1861, Riemann suggested that the function

is everywhere continuous, but nowhere differentiable in one of his lectures without proof. In order to give a formal proof of the Riemann function, Weierstrass discovered another everywhere continuous but nowhere differentiable function defined by where 0 < b < 1 and a is an integer greater than one. He proved that if \( ab > \left( 1 + \frac{3\pi}{2} \right) , \) the Weierstrass function, W(x) is nowhere differentiable. In 1916, Hardy proved that R(x) is not differentiable at all irrational multiples of π and at some rational multiples of π. In 1969, Joseph Gerver proved  that  the Riemann function is differentiable at all rational multiples of π of the form (πp/q), where p and q are odd integers and is not differentiable at other points. In the context of Fourier series, there is another function where 0 < α < 1. This is another continuous but nowhere differentiable function with many vanishing Fourier coefficients. A Fourier series with many missing Fourier coefficients, like W(x) and f(x) is called lacunary Fourier series. It can be shown that the real as well as the imaginary part of f(x) are both nowhere differentiable.

With ideas from Fourier series, Hermann Weyl (1885--1955) proved his famous criterion which states that a sequence of real numbers \( \left\{ x_n \right\}_1^{\infty} \) in [0, 1] is equidistributed for all integers k ≠ 0 if and only if

This means that, to understand the equidistribution properties of a real sequence \( \left\{ x_n \right\}_1^{\infty} , \) it is sufficient to estimate the size of the corresponding Fourier sum \( \sum_{n=1}^N \exp \left\{ 2\pi {\bf j} k\,x_n \right\} . \) For example, it can be shown using the Weyl criterion that the sequence of the fractional parts \( \left\langle n^2 \gamma \right\rangle = n^2 \gamma -\left\lfloor n^2 \gamma \right\rfloor , \) where ⌊A⌋ is the greatest integer ≤A, and is the integral part of n²γ and γ is irrational, is equidistributed in [0,1]. However, it can be proved that the sequence \( \left\{ x_n \right\}_1^{\infty} , \) where x_n is the fractional part of gⁿ, where \( g = \left( 1 + \sqrt{5} \right) /2 \) is the golden ratio, is not equidistributed in [0, 1].

As a remarkable application of the theory of finite Fourier series, we state Dirichlet’s theorem on primes in arithmetic progression. If a and h are positive integers with no common factor, then the arithmetic progression

contains infinitely many prime numbers. In this theorem, it is the theory of Fourier series on finite Abelian group ℤ^*(a) that plays a fundamental role in the solution of the problem, where ℤ^*(a) is defined as the set of all integers modulo a that have multiplicative inverses with multiplication modulo a as the group operation.

In 1924, Werner Heisenberg (1901--1976) first formulated his celebrated uncertainty principle between the position and momentum of a moving particle in quantum mechanics which states that it is not possible to determine the position and momentum of a particle exactly and simultaneously. This principle has an important physical interpretation as an uncertainty of both position and momentum of a particle described by a wave function ψ ∈ 𝔏²(ℝ), so it is a square-integrable function. Mathematically, this principle can be formulated in terms of a function and its Fourier transform. If f ∈ 𝔏²(ℝ), then f and its Fourier transform cannot both be essentially localized at x=0, k=0. In other words, it is not possible that the widths of the graphs of \( | f(x) |^2 \) and the square of its Fourier transform \( | F(k) |^2 \) can both be made arbitrarily small. This fact underlines the Heisenberg uncertainty principle in quantum mechanics and the bandwidth theorem in signal analysis. All these can be explained more explicitly as follows:

The function \( f_X (x) = |\psi (x)|^2 \) can be interpreted as the probability density for the random position variable X with its probabilistic interpretation

Similarly, \( f_P (k) = |{\cal F}\left\{ \psi (x) \right\} |^2 = \left\vert \hat{\psi} \right\vert^2 \) is the corresponding probability density function for its momentum P. The Heisenberg uncertainty principle states in the form of an exact inequality that these two probability densities cannot be essentially localized.

We then introduce the expectation of the random variable X² by

This can be regarded as a measure of the horizontal spread of the wave function ψ(x). Similarly, the spread of its Fourier transform, \( \hat{\psi} (k) \) over the k-axis can be introduced by the integral In terms of these two quantities, the uncertainty principle can be formulated in the form of a precise inequality, known as the Heisenberg inequality, where the equality holds when ψ(x) is the Gaussian function, that is, \( \psi (x) = C\,\exp \left\{ -a x^2 \right\} , \quad a> 0, \) and C is a constant.

It is important to point out new and modern impact of Fourier analysis on mathematical economics and finance. Based on the principle of Brownian motion for random fluctuations of share prices, Black and Scholes first formulated a dynamic replication model of price and show that the option price function, C(s, t) satisfies a partial differential equation with an extra drift term in the form

with the final boundary condition, C(s, t) = f(s), where r and σ² are constants. This equation is now known as the Black--Scholes (or BS) equation (see section in this tutorial). With a suitable change of variables, the BS equation can be reduced to the one-dimensional Fourier diffusion equation. Thus, the Fourier transform of C(s, t) with respect to s can be used to find solution of the BS model equation. Although the BS dynamic replication approach satisfies the principle of market completeness, one problem of this approach is that the price of a derivative does not depend on the drift of the share price. So, there is a need for an alternative approach to derivatives pricing theory, known as the risk-neutral expectation pricing theory which has also subsequently been developed. Thus, there are two major methods for pricing: the BS replication approach and the risk-neutral expectation approach. Surprisingly, the Feynman--Kac formula describing the relation between the solution of the Fourier diffusion equation and probability theory helped combine the above two approaches together so that certain second-order linear partial differential equation can be solved using expectations of diffusive processes. However, there is considerable evidence that logarithm of the share price does not follow the principle of Brownian motion with drift. In particular, when stock market crashes that corresponds to a big jump in the share price leading to a failure of the BS replication strategics. So, the BS model fails to capture some important features of share-price evolution because all paths in Brownian motion are continuous but non-differentiable. This leads to many generalization of the BS model, but most generalizations do not retain the principle of market completeness which means that they give rise to multiple option prices rather than just one price. The analysis of real market data on small-scale share movements reveals that they do not resemble a diffusion process. In fact, they behave more or less like a series of jumps than a Brownian motion. So, rescaling time based on the number of trades that have occurred rather than on calendar time leads to returns that do become approximately normal distribution. Thus, the BS model can be generalized based on trading time. An example of such a model is known as the variance gamma model. In general, the theory of Levy processes can be employed to develop more general theories of price movements for shares and other assets.

Since Fourier’s 1822 celebrated work on the theory of heat, the sine and cosine functions have played the fundamental basis functions in the subsequent development of Fourier series and Fourier transforms. However, in many applications, especially in the space-wavenumber (or time-frequency) analysis of a function (or signal), the standard Fourier analysis is not adequate because the Fourier transform of the function (or signal) does not contain any local information. In fact, it excludes the idea of wavenumbers changing with space (or frequencies changing with time), or equivalently, the notion of finding the wavenumber spectrum of a function with space (or frequency spectrum of a signal with time). So, the Fourier transform analysis of functions cannot be used for the analysis of functions (or signals) in a joint space wavenumber (or joint time-frequency) domain. So, in 1980s, a whole new idea of wavelets has been introduced as a family of functions constructed from translations and dilations of a single function called the ‘mother wavelet’ ψ(x). They are defined by

where a is called a scaling parameter which measures the degree of compression or scale, and b is a translation parameter which determines the space location of a wavelet. If ψ ∈ 𝔏² (ℝ) and ψ_a,b(x) is given by the latter equation, the continuous wavelet transform is defined by where ψ_a,b(x) plays the same role as the kernel exp(jkx) in the Fourier transform. Similar the Fourier transform, the continuous wavelet transform is linear, and it is not a single transform, but any transform obtained in this way. The inverse wavelet transform can be defined so that f(x) can be reconstructed by a double integral formula. It is now well-known that, analogous to the Fourier analysis, the wavelet analysis has provided a famous new method for decomposing a function or a signal in 1980s. It is more useful and better suited to a wide variety of problems in applied and computational mathematics and also to many questions in real and harmonic analysis. The most significant contributions of both discrete and continuous wavelets can be found in signal processing and digital image compression.

 

Historically, in 1747, d’Alembert first formulated the wave equation with the first solution of a partial differential equation. In fact, Fourier series originated from the early study of the initial-boundary problem of the one-dimensional wave equation for vibration of an elastic string fixed at its ends in the eighteenth century and of the Fourier heat equation of a solid body in 1800s. Euler, d’Alembert and Daniel Bernoulli made earliest attempts to solve the initial-boundary value problem of the classical one-dimensional wave equation

where u = u(x,t) is the displacement of the string of length ℓ at the position x and time t, c² = (T/ρ), T is the tension and ρ s the mass density of the string.

The initial and boundary conditions of this problem are given by

where d(x) is the initial displacement and v(x) is the initial velocity of the string.

Both Euler and d’Alembert made the first attempt to solve the initial value problem as the traveling wave solution in the functional form

where φ(x-ct) represents a wave traveling to the right with constant velocity c and φ(x+ct) represents a wave moving to the left with constant speed c. The above solution is the general solution of the wave equation representing the superposition of two traveling wave solutions.

On the other hand, Daniel Bernoulli derived the solution of the wave equation with the boundary conditions satisfied in an infinite trigonometric series form

where a_n and b_n are to be determined from the initial conditions This means that "arbitrary" functions d(x) and v(x) have been expanded in trigonometric (or Fourier) series. Since the above equations are infinite series, the question of convergence arose. But many mathematicians including Euler, d’Alembert and Laplace were not entirely convinced about the existence of Bernoulli’s Fourier series solution. This led to a serious controversy as the solution could hold if an arbitrary function could be represented in Fourier series. In 1759, a young mathematical scientist, Lagrange joined in this controversial debate on the various forms of the solution of the vibrating string problem and published a paper which was criticized by both Euler and d’Alembert. In order to resolve this controversy, Lagrange gave an entirely new treatment of the problem. From 1768, d’Alembert wrote a series of notes in which he attacked Euler’s views on the solution of the problem. In 1779, Laplace entered into this debate and supported d’Alembert views. They all discovered solutions in several different forms and the merit of their solutions and their relations among these solutions extending over more than 25 years. Their major concerns were whether an arbitrary function can be expanded in Fourier series and convergence of such series. Unfortunately, the outcome of this controversial issues remained inconclusive for a long period of time.

In 1807, it was Joseph Fourier who resolved this long-standing controversy by discovering the solution of the one-dimensional heat equation in terms of Fourier series. Indeed, he formulated and solved the initial-boundary value of the one-dimensional heat equation in a rod of length

with the initial and boundary conditions Using the method of separation of variables, Fourier obtained the most general and rapidly convergent solution for the temperature distribution u(x, t) in the form where the initial condition implies that The upshot of Fourier’s analysis led to expanding an arbitrary function in an infinite series of sine functions and then finding the infinite number of coefficients

 

References

 

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