Fourier series

“ Heine showed (Crelle’s Journal, vol. lxxi., 1870, and in his treatise Kugelfunctionen, vol. i.) that Fourier’s Series is in general uniformly convergent, and that if there is a uniformly convergent series which represents a function, it is the only one of the kind. G. Cantor then showed (Crelle’s Journal, vols. lxxii. lxxiii.) that even if uniform convergence be not demanded, there can be but one convergent expansion for a function, and that it is that of Fourier. In the Math. Ann. vol. v., Cantor extended his investigation to functions having an infinite number of discontinuities. ”

“ A memoir by Brodén (Math. Annalen, vol. lii.) contains a good investigation of some of the most recent results on the subject. The scope of Fourier’s Series has been extended by Lebesgue, who introduced a conception of integration wider than that due to Riemann. Lebesgue’s work on Fourier’s Series will be found in his treatise, Leçons sur les séries trigonométriques (1906) ; also in a memoir, “Sur les séries trigonométriques,” Annales sc. de l’école normale supérieure, series ii. vol. xx. (1903) , and in a paper “Sur la convergence des séries de Fourier,” Math. ”

“ Upon the banks of the Nile, Fourier devoted himself to assiduous researches on almost every branch of knowledge which the vast plan of the Institute embraced. The Decade and the Courier of Egypt will acquaint the reader with the titles of his different labours. I find in these journals a memoir upon the general solution of algebraic equations; researches on the methods of elimination; the demonstration of a new theorem of algebra; a memoir upon the indeterminate analysis; studies on general mechanics ”

“ Functions have also been considered which for an infinite number of points within the given interval have no definite value, and series have also been discussed which at an infinite number of points in the interval cease either to have a sum, or to have one which agrees with the value of the function; the narrower conception above will however be retained in the treatment of the subject in this article, reference to the wider class of cases being made only in connexion with the history of the theory of Fourier’s Series. ”

“ The Ultimate Values of the Coefficients in Fourier’s Series.—If ƒ (x) is everywhere finite within the given interval −π to +π, it can be shown that an, bn, the coefficients of cos nx, sin nx in the series which represent the function, are such that nan, nbn, however great n is, are each less than a fixed finite quantity. ”

“ A scientific subject does not occupy so much space in the life of a man of science of the first rank without being important and difficult. The subject of algebraic analysis above mentioned, which Fourier had studied with a perseverance so remarkable, is not an exception to this rule. It offers itself in a great number of applications of calculation to the movements of the heavenly bodies, or to the physics of terrestrial bodies, and in general in the problems which lead to equations of a high degree. ”

“ Now Fourier gives no demonstration of the possibility of this expansion, but he merely determines what the coefficients , , & would be, if the function were represented by a series of this form. Poisson arrives, by another method, at the same conclusion as Fourier, and then states this objection to Fourier’s solution; but, as is remarked by Mr Kelland (Theory of Heat, p. 81, note) , he “does not appear, as far as I can see, to get over the difficulty.” ”

“ Since the curve represents a longitudinal disturbance in air it is always continuous, at a finite distance from the axis, and with only one ordinate for each abscissa.
Fourier's theorem asserts that such a curve may be built up by the superposition, or addition of ordinates, of a series of sine curves of wave-lengths X, JX, JX, fX. . . if the amplitudes a, b, c. . .and the epochs e, f, g. . . . are suitably adjusted, and the proof of the theorem gives rules for finding these quantities when the original curve is known. ”

“ I shall add even the reputation of a professor full of grace, for our colleague has proved that this kind of merit may not be foreign to the teaching of mathematics.
The lectures of Fourier have not been collected together. The Journal of the Polytechnic School contains only one paper by him, a memoir upon the "principle of virtual velocities." ”

“ Restored at length, after so many vicissitudes, to his favourite pursuits, Fourier passed the last years of his life in retirement and in the discharge of academic duties. To converse had become the half of his existence. Those who have been disposed to consider this the subject of just reproach, have no doubt forgotten that constant reflection is no less imperiously forbidden to man than the abuse of physical powers. Repose, in every thing, recruits our frail machine ”

“ He accordingly shows himself deaf to the clamours of party, and Fourier receives from him the superior direction of the Bureau de la Statistique of the Seine, with a salary of 6,000 francs. It has appeared to me, Gentlemen, that I ought not to suppress these details. Science may show herself grateful towards all those who give her support and protection, when there is some danger in doing so, without fearing that the burden should ever become too heavy. ”

“ Labours relative to the Solar System. Optical Labours. LAPLACE. Preliminary Notice. APPENDIX. (A.) Brief Notice of some other interesting Results of the Researches of Laplace which have not been mentioned in the Text. (B.) The Mécanique Céleste. JOSEPH FOURIER. Preliminary Notice. Birth of Fourier.—His Youth. Memoir on the Resolution of Numerical Equations. Part played by Fourier in our Revolution.—His Entrance into the Corps of Professors of the Normal School and the Polytechnic School.—Expedition to Egypt. Fourier Prefect of L'Isère. Mathematical Theory of Heat. ”

“ At a later period the process received a brilliant extension at the hands of Maxwell, in his theory of gases, where it was applied to the flux of momentum and also of energy.
The mathematical methods employed by Fourier in his treatment of special problems repay a careful study. As they stand they would often fail to satisfy even a lenient standard of mathematical rigour, and indeed they appear to have raised doubts in the minds of Laplace, [Pg 12] Lagrange, and Legendre, who formed the distinguished commission charged to examine one of his memoirs. ”

“ A function which has a finite differential coefficient at all points of an interval is continuous throughout the interval, but if the differential coefficient becomes infinite at a point of the interval the function may or may not be continuous throughout the interval; on the other hand a function may be continuous without being differentiable. This result, comparable in importance, from the point of view of the general theory of functions, with the discovery of Fourier’s theorem, is due to G. F. B. Riemann ”

“ We proceed to the forced vibrations of the system. The typical case is where the extraneous forces are of the simple-harmonic type cos (σt + ε) ; the most general law of variation with time can be derived from this by superposition, in virtue of Fourier’s theorem. Analytically, it is convenient to put Qr, equal to eiσt multiplied by a complex coefficient; owing to the linearity of the equations the factor eiσt will run through them all, and need not always be exhibited. ”

“ Here is zero or unity according as is even or odd, and although does not become infinite as increases without limit, it does not tend to a limit, but oscillates. It is evident that if all the terms of a series have the same sign, the series cannot oscillate.
Since the sum of a converging series is a perfectly definite number, while such a thing as the sum of a nonconvergent series does not exist, it follows at once that it is absolutely essential in any given problem involving infinite series to determine whether or not the series is convergent. ”

“ The irreversible character of this process is strikingly embodied in Fourier's theory of the conduction of heat, where the formulae themselves indicate, for all positive values of the time, a possible solution which continually tends to the form of a uniform diffusion of heat.
But if we attempt to ascend the stream of time by giving to its symbol continually diminishing values, we are led up to a state of things in which the formula has what is called a critical value; and if we inquire into the state of things the instant before, we find that the formula becomes absurd. ”

“ I do not profess to judge of the merits of the case, but one feature distinguishes Fourier from critics, reformers, and prophets, who are gathered to their fathers. He is a scientific explorer, and the plans that he has designed for the future structure of humanity, from the high order of architecture and mechanics which they exhibit, discriminate him from the vulgar herd as an originator, and place him in the category either of eminent scientific adventurers or inventors. Daring and caution shake hands at every page, and seem exhausted by his pen. ”

“ The next ten papers, with two exceptions, are all on the motion of heat, and appeared in the Cambridge Mathematical Journal between 1841 and 1843, and deal with important topics suggested by Fourier's treatise. Of the ideas contained in one or two of them some account will be given presently.
Fourier's book was called by Clerk Maxwell, himself a man of much spirituality of feeling, and no mean poet, a great mathematical poem. Thomson often referred to it in similar terms. The idea of the mathematician as poet may seem strange to some ”

“ The investigation of the mathematical phenomena of the laws of heat by Baron Fourier has led to the establishment, in an entirely direct manner, of Thermological equations. This great discovery tends to elevate our philosophical hopes as to the future extensions of the legitimate applications of mathematical analysis, and renders it proper, in the opinion of author, to regard Thermology as a third principal branch of concrete mathematics. ”

“ It was a great improvement when the consideration of molecular forces could be dispensed with and replaced by Cauchy’s theory of stress. The same tendency to discard unnecessary and unverifiable hypothesis has been exemplified in Electricity, in the transition from Poisson and Ampère to Thomson and Maxwell.
One feature which is met with in our period is the frank appeal to intuition. This is noticeable already in the case of Fourier, as has been already indicated, but it runs through the whole school. ”

“ Fourier teaches us that this number is necessary, and if he has taught the truth of the science, it is worse than folly to pursue a course contrary to his instructions. If there is any one who understands the science better than Fourier did himself, we hope he will make the necessary corrections and send us word. ”

“ We may omit, for the time, from our notice, the before-mentioned possibility of a knowledge of the world that would reduce it to a single serial order of the general type of the Well-Ordered Series of whole numbers, where every term has one coming next after it. If we abstract from that possibility, we are left to the conception that Between any two facts there are to be found various series of intermediaries of the type now defined. The world thus regarded will consist for us of all these interwoven series, and will constitute a single System. ”

“ In the world of our pure conceptions, we find that between successive whole numbers, such as 2 and 3, 3 and 4, it is impossible to conceive other whole numbers inserted, so long as one takes the whole numbers in their natural order. And the same holds true, in the world of empirical things, regarding any simple series of objects whose type is that of the whole number series, where every object of the series is followed by a next one, with nothing between that belongs to the series. ”

“ Whatever any one may believe of the feasibility of the carrying out of Fourier's doctrines of united industry or the practicality of any of his theories, they must stand amazed at the bold and often extremely beautiful conceptions of his brain; such as the actual forecasting of the development theory before Darwin, Spencer and Huxley were born—though not exactly in detail with them; his bolder conception still of the destiny of man, and his Cosmogony ”

“ Despite the aberrations of Fourier, the lucid portions of his system survive and will bear a critical examination. Truth triumphs and pursues its way through whatever disguise one views it and in whatever disguise one clothes it. But it would be much better that, in the age of reason which we have reached, the ridiculous manifestations of a blind enthusiasm should disappear entirely. Is not that your opinion? Has not the hour struck when serious-minded people should take possession of their true domain, and when those things that are logically proved should be professed by logicians? ”

“ In other words, no part of the energy depends on the product of the amplitudes of two different harmonics. When two modes of motion of the same system are conjugate to each other, the existence of one of them does not affect the other.
The simplest case of harmonic analysis, that of which the treatment of the vibrating string is an example, is completely investigated in what is known as Fourier’s theorem. ”

“ Examining myself, therefore, upon this charge of Fourierism, and endeavoring to refresh my memory, I find that, having been connected with the Fourierists in my studies and my friendships, it is possible that, without knowing it, I have been one of Fourier’s partisans. Jérôme Lalande placed Napoleon and Jesus Christ in his catalogue of atheists. The Fourierists resemble this astronomer: if a man happens to find fault with the existing civilization, and to admit the truth of a few of their criticisms, they straightway enlist him, willy-nilly, in their school. ”