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Review Problems for the Midterm Examination
February 14, 2000
1.
Define an inner product of a linear space
V
that is a set of all continuous functions defined on an
interval
()
0,1 , and show that it satisfies the required properties of the inner product. Orthogonalize
three functions
() () ()
2
12
3
1,
,and
xx
x
φφ
φ
==
=
with respect the inner product you have
defined
2.
State the required properties of a norm
. in a linear space V. What is the natural norm? Show that
the following two norms satisfy the required properties of a norm:
[]
0,1
max
x
ff
x
∈
=
1
2
0
x
d
x
=
∫
where
V
is a set of all continuous functions defined on
0,1 . Show that the inequality
1
2
0
0,1
max
x
fx d
x
fx
∈
≤
∫
.
State your idea whether or not a positive constant
0
α
>
exists that satisfies the inequality
1
2
0
0,1
max
x
x
∈
≤
∫
.
3. Suppose that a data set
{}
,
1,.
..,
1
i
fi
n
=+
is given at a set of sampling points
,1
,
.
.
.
i
xi
n
. Define the least squares method to approximate a function
by using a
set of linearly independent functions
,
1,.
..,
1
k
xk
m
. Also find the necessary condition
of the least squares method. For the data
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 Fall '09

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