4 INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS In order to simplify the notations throughout, we recall the Euler formula for the complex exponential function elG = cos 0 + / sin 0 and its consequences 1 -£?*+e~w\ 1 sin6 = -(eie -e'w). 2/ (1.1.13) cos This allows us to rewrite the trigonometric series (1.1.12) in the complex form (1.1.14) f(0) = J2 C ^n6 Complex notation is especially efficient when we multiply two such functions, the for- mula eldel(f) = el(6+^ being a streamlined expression of the addition formulas for both the sine and cosine functions. Similar efficiency is realized in the integration formulas / eax dx a"xeax when a is a nonzero complex number. The passage from real functions to complex functions also suggests the natural definition of convergence of the series (1.1.14), namely as the limit of the symmetric partial sums YZN- ^ e u s e throughout the notation J2nez f°r t r n s limiting process. If we try to consider more general definitions of convergence, difficulties will arise. Throughout the text we will systematically use the Lebesgue integral and its many properties. In some cases a more elementary definition of integration will suffice, but we prefer to systematically employ the Lebesgue theory—both for its increased generality and its ease with respect to passage to the limit. Theorem 1.1.5. Suppose that Ylnei \^n\ °°- Thenf defined by (1.1.14) is a continuous function on T. The coefficients are obtained as (1.1.15) Ifg is any other Lx function on T, we have the Fourier reciprocity formula (1.1.16) where Dn is the Fourier coefficient of g, defined by (1.1.15) withf replaced by ± In particular we have ParsevaVs identity (1.1.17) Proof. The uniform convergence of (1.1.14) follows from the Weierstrass M test, since | Cnem91 = \Cn\, the general term of a convergent numerical series, so that the limit function is continuous. This uniform convergence also holds for the series defined by e~lN9f(0),
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