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Miklós Schweitzer
1995 Miklós Schweitzer
3
3
Part of
1995 Miklós Schweitzer
Problems
(1)
analysis
Source: miklos schweitzer 1995 q3
10/5/2021
Denote
⟨
x
⟩
\langle x\rangle
⟨
x
⟩
the distance of the real number x from the nearest integer. Let f be a linear, 1 periodic, continuous real function. Prove that there exist natural n and real numbers
a
1
,
.
.
.
,
a
n
,
b
1
,
.
.
.
,
b
n
,
c
1
,
.
.
.
,
c
n
a_1 , ..., a_n , b_1 , ..., b_n , c_1 , ..., c_n
a
1
,
...
,
a
n
,
b
1
,
...
,
b
n
,
c
1
,
...
,
c
n
such that
f
(
x
)
=
∑
i
=
1
n
c
i
⟨
a
i
x
+
b
i
⟩
f(x) = \sum_{i = 1}^n c_i \langle a_ix + b_i \rangle
f
(
x
)
=
i
=
1
∑
n
c
i
⟨
a
i
x
+
b
i
⟩
for every x iff there is a k such that
∑
j
=
1
2
k
f
(
x
+
j
2
k
)
\sum_{j = 1}^{2^k} f \left(x+{j\over2^k}\right)
j
=
1
∑
2
k
f
(
x
+
2
k
j
)
is constant.
real analysis