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Fourier Analysis in L1 Introduction

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Fourier Analysis in L1 Introduction Part A Fourier Analysis in L1 Introduction In 1807 Joseph Fourier (1768-1830) presented a solution ofthe heat equationl ae a2e -=K-, at a2x where e(x, t) is the temperature at time t and at loeation x of an infinite rod, and K is the heat eonduetanee. The initial temperature distribution at time 0 is given: e(x,O) = f(x). (The solution of the heat equation is derived in Seetion A 1·1.) In fact, Fourier eonsidered a cireular rod of length, say, 21T, whieh amounts to imposing that the funetions x --+ f(x) and x --+ e(x, t) are 21T-periodie. He gave the solution when the initial temperature distribution is a trigonometrie series f(t) = L cneint . neZ Fourier claimed that his solution was general beeause he was eonvineed that alI21T­ periodie funetions ean be expressed as a trigonometrie series with the eoefficients 1 2lT -int Cn = cn(f) = - 1 f(t)e dt. 21T 0 lThe definitive form of his work was published in Theorie Analytique de la Chaleur, Finnin Didot ed., Paris, 1822. 4 Part A Fourier Analysis in L I Special cases of trigonometric developments were known, for instance, by Leonhard Euler (1707-1783), who gave the formula 1 . 1 . 1 . "2 x = sm(x) - "2 sm(2x) + 3" sm(3x) - "', true for -l( < x < +l(. But the mathematicians of that time were skeptical about Fourier's general conjecture. Nevertheless, when the propagation of heat in solids was set as the topic for the 1811 annual prize of the French Academy of Sciences, they surmounted their doubts and attributed the prize to Fourier's memoir, with the explicit mention, however, that it lacked rigor. Fourier's results that were in any case true for an initial temperature distribution that is a finite trigonometric sum, and be it only for this, Fourier fully deserved the prize, because his proof uses the general tricks (for instance, the differentiation rule and the convolution-multiplication rule) that constitute the powerful toolkit of Fourier analysis. Nevertheless, the mathematical problem that Fourier raised was still pending, and it took a few years before Peter Gustav Dirichlet2 could prove rigorously, in 1829, the validity of Fourier's development for a large class of periodic func­ tions. Since then, perhaps the main guideline of research in analysis has been the consolidation of Fourier's ingenious intuition. The classical era of Fourier series and Fourier transforms is the time when the mathematicians addressed the basic question, namely, what are the functions adrnit­ ting a representation as a Fourier series? In 1873 Paul Dubois-Reymond exhibited a continuous periodic function whose Fourier series diverges at O. For almost one century the threat of painful negative results had been looming above the theory. Of course, there were important positive results: Ulisse Dini3 showed in 1880 that if the function is locally Lipschitz, for instance differentiable, the Fourier series represents the function. In 1881, Carnille Jordan4 proved that this is also true for functions of locally bounded variation. Finally, in 1904 Leopold Fejeii showed that one could reconstruct any continuous periodic function from its Fourier coef­ ficients. These results are reassuring, and for the purpose of applications to signal processing, they are sufficient. However, for a pure mathematician, the itch persisted. There were more and more examples of periodic continuous functions with a Fourier series that diverges at at least one point. On the other hand, Fejer had proven that if convergence is taken in the Cesaro sense, the Fourier series of such continuous periodic function converges to the function at all points. 2Sur la convergence des series trigonometriques qui servent a representer une fonetion arbitraire entre des limites donnees, J. reine und angewan. Math., 4,157-169. 3 Serie di Fourier e altre rappresentazioni analitiche delle funzioni di une variabile reale, Pisa, Nistri, vi + 329 p. 4Sur la serie de Fourier, CRAS Paris, 92, 228-230; See also Cours d'Analyse de l'lfcole Polytechnique, I, 2nd ed., 1893, p. 99. 5Untersuchungen über Fouriersehe Reihen, Math. Ann., 51-69. Introduction 5 Outside continuity, the hope for a reasonable theory seemed to be completely destroyed by Nikola'i Kolmogorov,6 who proved in 1926 the existence of a periodic locally Lebesgue-integrable function whose Fourier series diverges at alt points! It was feared that even continuity could foster the worst pathologies. In 1966 Jean­ Pierre Kahane and yitzhak Katznelson 7 showed that given any set of null Lebesgue measure, there exists a continuous periodie function whose Fourier series diverges at all points of this preselected set. The case of continuous functions was far from being elucidated when Lennart Carleson8 published in the same year an unexpected result: Every periodic locally square-integrable function has an almost-everywhere convergent Fourier series. This is far more general than what the optirnistic party expected, since the periodic continuous functions are, in particular, locally square-integrable. This, together with the Kahane-Katznelson result, completely settled the case of continuous periodic functions, and the situation finally tumed out to be not as bad as the 1873 result of Dubois-Reymond seemed to forecast. In this book, the reader will not have to make her or his way through a jungle of subtle and difficult results. Indeed, for the traveler with practical interests, there is a path through mathematics leading directly to applications. One of the most beautiful sights along this road may be Simeon Denis Poisson's9 sum formula L J(n) = L jen), nEZ nEZ where J is an integrable function (satisfying some additional conditions to be made precise in the main text) and where j(v) = LJ(t)e-2irrvt dt is its Fourier transform, where ~ is the set of real numbers. This striking formula found very nice applications in the theory of series and is one of the theoretical results founding signal analysis. The Poisson sum formula is the culrninating result of Part A, which is devoted to the classical Fourier theory. 6Une serie de Fourier-Lebesgue divergente partout, CRAS Paris, 183, 1327-1328. 7Sur les ensembles de divergence des series trigonometriques, Studia Mathematica, 26, 305-306. 8Convergence and growth of partial sums of Fourier series, Acta Math., 116, 135-157. 9Sur la distribution de la chaleur dans les corps solides, J. Ecole Polytechnique, 1geme Cahier, XII, 1-144, 145-162. .
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