FOURIER SERIES ON THE CIRCLE 5 which we may therefore integrate term-by-term. In the process we encounter the integral (1.1.18) f e-iNeeinedO = 0 (n ^ N,n e Z,N e Z) J-n while the integral is 2TC for n = N. Equation (1.1.18) is known as the complex orthogonality relation. This shows that (1.1.19) i - j " e-iN6f(0)dO = CN, (N e Z) which was to be proved. To prove (1.1.16), we multiply the series (1.1.14) by g(0). The partial sums are bounded by a multiple of the integrable function \g(0)\, hence we can apply Lebesgue's dominated convergence and integrate term-by-term to obtain (1.1.20) -L I" f (0)8(0) d0 = T ^ r eineg(0)d0 = ~L Vc„D_„ which was to be proved. Taking g = f gives the Parseval identity (1.1.17). Exercise 1.1.6. Suppose that 5Znez \n^n\ oo. Prove that the series (1.1.14) defines a differentiable function with f (9) = Ylnez inCneine a continuous function. Hint: Use the inequality \elh 1| \h\ to justify passage to the limit. Exercise 1.1.7. Suppose that ^ e Z \nkCn\ oo for some k = 2, 3, Prove that the series (1.1.14) defines a k-times differentiable function with f^k\0) = ^n^i(iri)kCnemd a continuous function. The Fourier reciprocity formula (1.1.16) can be rewritten to obtain a useful rep- resentation of the convolution of an absolutely convergent trigonometric series with an arbitrary integrable function. Taking g^iO) = g{(j) 0), we compute the Fourier coefficient by writing [ e-in6g^0) d0= [ eW'+lgW) df = e~in* [ g{f)ein* df which, when substituted into (1.1.16), yields the following. Corollary 1.1.8. The convolution of an absolutely convergent trigonometric series f with an arbitrary L1 function g has the representation (1.1.21) A large source of examples of absolutely convergent trigonometric series is obtained from power series. Consider a general Laurent series oo oo (1.1.22) f(z) = J2anZ" + Y,b"z~n' n=0 n=\
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