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Putnam
1993 Putnam
B4
Putnam 1993 B4
Putnam 1993 B4
Source: 1993 Putnam
October 26, 2020
Putnam
real analysis
Problem Statement
K
(
x
,
y
)
,
f
(
x
)
K(x, y), f(x)
K
(
x
,
y
)
,
f
(
x
)
and
g
(
x
)
g(x)
g
(
x
)
are positive and continuous for
x
,
y
⊆
[
0
,
1
]
x, y \subseteq [0, 1]
x
,
y
⊆
[
0
,
1
]
.
∫
0
1
f
(
y
)
K
(
x
,
y
)
d
y
=
g
(
x
)
\int_{0}^{1} f(y) K(x, y) dy = g(x)
∫
0
1
f
(
y
)
K
(
x
,
y
)
d
y
=
g
(
x
)
and
∫
0
1
g
(
y
)
K
(
x
,
y
)
d
y
=
f
(
x
)
\int_{0}^{1} g(y) K(x, y) dy = f(x)
∫
0
1
g
(
y
)
K
(
x
,
y
)
d
y
=
f
(
x
)
for all
x
⊆
[
0
,
1
]
x \subseteq [0, 1]
x
⊆
[
0
,
1
]
. Show that
f
=
g
f = g
f
=
g
on
[
0
,
1
]
[0, 1]
[
0
,
1
]
.
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