20. The following results are relevant in information theory when one tries to recover a signal...

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20. The following results are relevant in information theory when one tries to recover a signal from its samples. Suppose f is of moderate decrease and that its Fourier transform is sup- ported in I = [1/2, 1/2]. Then, f is entirely determined by its restriction to Z. This means that if g is another function of moderate decrease whose Fourier transform is supported in I and f(n) = g(n) for all n Z, then = g. More precisely: (a) Prove that the following reconstruction formula holds: sin Ty f(x) = f(n)K(x n) - where K(y) = Note that K(y) = O(1/|y|) as |y| . (b) If > 1, then f(x) = (1) *. (*-*) } } Kx cos where K(y) COS TY - COSTy = ( 1)y - Thus, if one samples f "more often," the series in the reconstruction formula converges faster since Kx(y) = O(1/|y|) as y |y| . Note that K(y) K(y) as 1. (c) Prove that [ |f(x)| dx = \(n)|. N=1 [Hint: For part (a) show that if x is the characteristic function of I, then () = x() x_ f(n)e-2ing. For (b) use the function in Figure 2 instead of x().] n= - 20. The following results are relevant in information theory when one tries to recover a signal from its samples. Suppose f is of moderate decrease and that its Fourier transform is sup- ported in I = [1/2, 1/2]. Then, f is entirely determined by its restriction to Z. This means that if g is another function of moderate decrease whose Fourier transform is supported in I and f(n) = g(n) for all n Z, then = g. More precisely: (a) Prove that the following reconstruction formula holds: sin Ty f(x) = f(n)K(x n) - where K(y) = Note that K(y) = O(1/|y|) as |y| . (b) If > 1, then f(x) = (1) *. (*-*) } } Kx cos where K(y) COS TY - COSTy = ( 1)y - Thus, if one samples f "more often," the series in the reconstruction formula converges faster since Kx(y) = O(1/|y|) as y |y| . Note that K(y) K(y) as 1. (c) Prove that [ |f(x)| dx = \(n)|. N=1 [Hint: For part (a) show that if x is the characteristic function of I, then () = x() x_ f(n)e-2ing. For (b) use the function in Figure 2 instead of x().] n= -