2 INTRODUCTION TO FOURIER ANALYSIS AND WAVELETS f(x, 0) = sin3x = (3sinx sin3je)/4, whereas sin2x cannot be so expressed. In order to work with these trigonometric sums, we note the property of orthogonality, expressed as r 1 sin mx sin nx dx = 0, Jo m 7^ n. If a function has the form f{x) = Ylk=\ a ksinfac, m e n w e must have JQ fix) sin nx dx = 0 for n N. Exercise 1.1.1. Show that ifN is odd, sinN x can be written as a finite sum of the form Ylk=\ a k s m kx. Exercise 1.1.2. Suppose that we have a convergent series expansion of sin x = X^fcli a k sin be on the interval 0 x n. Prove that a^ is nonzero for infinitely many values ofk. Hint: Assume a finite expansion and use the orthogonality relation (1.1.3) to obtain a contradiction. Exercise 1.1.3. Generalize Exercise 1.1.2 to any even power of sin x, showing the impossibility of an expansion sin"* = J2k=\ a k sin be/or 0 x n where n = 4, 6, Any multiple harmonic motion (1.1.2) is a 2TC -periodic function of time:/(jc, t + 2n) = f(x, t) for all oc JC, oo, oo t oo. It also is a 2n-periodic function of x and is odd with respect to x = 0 and x = n, meaning that/(—JC) = —fix) and fin +x) = —fin x) for all x. Exercise 1.1.4. Suppose that fix), —oo x oo is given. Show that any two of the following properties imply the third: (i)fix + 2n) = fix), Vx (ii)f(-x) = —fix), Wx (iii)fin x) =—fin-\-x), VJC. /. /. 1.2 Heat flow in solids The vibrating string suggests the use of sine series, since the ends of the string are fixed. More general trigonometric series are suggested by the study of heat flow in a circular ring, assumed to have circumference 2n. In this model it is natural to assume that the temperature uix, t) is a 2n-periodic function of x (but not periodic in time). Fourier (1822) formulated the heat equation ut = uxx to describe the time evolution of the temperature. It is satisfied by any function of the form (An cos nx + Bn sin nx)e~n t where n = 0, 1, 2, ... , t 0 and n x n. Taking linear combinations of these, we arrive at a "general solution" TV (1.1.4) uix, t) = Y^(A„cosn c + Bnsinnx)e~n r . n=0
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