Chapter 1 Fourier Series and Integrals 1. Fourier coefficients and series The problem of representing a function / , defined on (an interval of) R, by a trigonometric series of the form oo (1.1) f(x) = ^ P ak cos(kx) + bk sin(kx) k=o arises naturally when using the method of separation of variables to solve partial differential equations. This is how J. Fourier arrived at the problem, and he devoted the better part of his Theorie Analytique de la Chaleur (1822, results first presented to the Institute de France in 1807) to it. Even earlier, in the middle of the 18th century, Daniel Bernoulli had stated it while trying to solve the problem of a vibrating string, and the formula for the coefficients appeared in an article by L. Euler in 1777. The right-hand side of (1.1) is a periodic function with period 27r, so / must also have this property. Therefore it will suffice to consider / on an interval of length 2TT. Using Euler's identity, elkx = cos(kx) + isin(kx), we can replace the functions sin(kx) and cos(kx) in (1.1) by {elkx : k G Z} we will do so from now on. Moreover, we will consider functions with period 1 instead of 27r, SO we will modify the system of functions to {e2nikx : k G Z}. Our problem is thus transformed into studying the representation of / by oo (1.2) f{x) = J2 Cke2*ikx. k=—oo l
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