 # 1. An isometry between the Hilbert space H of square integrable fun...

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1. An isometry between the Hilbert space H of square integrable functions f, and the Hilbert space l2 of square summable sequences {ch} is a linear one-to-one and onto transformation f {ck} with the property that it preserves squared lengths: 8 (f.f) Elail, = , Vf E H . k=1 WITHOUT assuming the existence of an orthonormal basis, SHOW that 8 = , Vf,gEH k=1 where f {Ck} and g {dk}. 2. Consider each of the following two sets: (a) The set of all (x1,22, , xk, ) € e2 such that x1 x2; (b) The set of all (x1,22, ,XK, ) € 2² such that Xk = 0 for all even k. TRUE or FALSE? Each of these two sets is a subspace of the Hilbert space l2. If TRUE, prove it. If FALSE, why not? Nota bene: Sets such as these represent, among others, data acquired from audio, mechanical, and other systems. 3. Let g be a fixed and given square integrable function, i.e. 8 0 < g(x)g(x) da ||g||² <00 FOO The function g must vanish as x 00. Consequently, one can think of g as a function whose non-zero values are concentrated in a small set around the origin x = 0. Consider the concomitant "windowed" Fourier transform on L² (-00,00), the space of square integrable functions, T: [2(-00,00) R(T) 00 f Tf(w,t) - Let h(w,t) be an element of the range space R(T) . It is evident that 8 00 = h1(w,t)h2(w,t) dwd -00 - 80 is an inner product on R(T). FIND a formula for (Tf1, Tf2) in terms of the inner product 8 (f1,f2) IIIE f1 (x) f2 (x) da on L2(-00,00). Nota bene: In this problem, which encourages you to jump ahead in your reading, you can use the result x 00 W - 00 x'=00 - = 2nt((c) - x) h(x) d. x - 00 W 00 x 00 which will need to be justified. 4. Suppose that f(x + 2n) = f(x) is an integrable function of period 27. SHOW that 2nta 27. f(x)dx = f(x) da a where a is any real number

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