2 1. Fourier Series and Integrals If we assume, for example, that the series converges uniformly, then by multiplying by e ~2mmx and integrating term-by-term on (0,1) we get Jo x )e-2nimxdx because of the orthogonality relationship (1.3) f1 e^ikxe-2*imxdx={® {ik ^m Jo 1 if k = m. Denote the additive group of the reals modulo 1 (that is 1R/Z) by T, the one-dimensional torus. This can also be identified with the unit circle, S1. Saying that a function is defined on T is equivalent to saying that it is defined on R and has period 1. To each function / G L1(T) we associate the sequence {/(&)} of Fourier coefficients of / , defined by (1.4) f(k)= [ f(x)e-2*ikxdx. Jo The trigonometric series with these coefficients, . 2ivikx (1-5) £ /(*) k=—oo is called the Fourier series of / . Our problem now consists in determining when and in what sense the series (1.5) represents the function / . 2. Criteria for pointwise convergence Denote the iV-th symmetric partial sum of the series (1.5) by SNI(X)] that is, N s N f(x) = Y, f(ky^nikx k=-N Note that this is also the iV-th partial sum of the series when it is written in the form of (1.1). Our first approach to the problem of representing / by its Fourier series is to determine whether limSjv/(aO exists for each x, and if so, whether it is equal to /(#). The first positive result is due to P. G. L. Dirichlet (1829), who proved the following convergence criterion: if / is bounded, piecewise continuous, and has a finite number of maxima and minima, then lim Spif(x) exists and is equal to \[f(x+) + f(x—)]. Jordan's criterion, which we prove below, includes this result as a special case.
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