 # 1. Consider the set of functions (&quot;wave packets&quot;)...

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1. Consider the set of functions ("wave packets") (j+ 1) € 1 { Pj1(t) = 1 e - iwt dw; j = 0, 1, 2, € 27 l = 0, 1, 2, je where € is a fixed positive constant. (a) SHOW that these wave packets are orthonormal: 90 i.e. Pj1(t) Pj'V (t) dt = Sizidiv -80 2. The convolution of two functions f and g is defined to be the new function f * g 8 f * g (x) f(x-t)g(t) dt - (**) whenever the integral converges. (b) If F(k) = ito exp (-ikx) f(x) dac and G(k) = dac are the Fourier transforms of f (x) and g(x), FIND the Fourier transform of h(x) = f * g (x). . (c) Is it true that (f * g) * h (x) = f * (g * h) (x) ? If yes, can this Associativity Law be validated with the help of (b)? 3. The Fourier transform, call it F , is a linear one-to-one operator from the space of square-integrable functions onto itself. (In fact, we also know that F is an "isometric" mapping, but we will not need this feature in this problem). Indeed, F: [2(-00,00) [2(-00,00) - 1 f(x) F(f)(k) e-ika f (x) da F(k) 27 -00 Note that here x and k are viewed as points on the common domain (-x0,00) of f and F. (a) Consider the linear operator F2 and its eigenvalue equation F2f=Xf. = What are the eigenvalues and the eigenfunctions of F2 ? (b) Identify the operator F4? What are its eigenvalues? 4. Express 6(xa) in terms of the Dirac delta "function" (x), , where a is a non-zero real constant.

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