Transcribed Text
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 onetoone 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 g² <00
FOO
The function g must vanish as x
00. Consequently, one can think of g as a function whose
nonzero 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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