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Miklós Schweitzer
1963 Miklós Schweitzer
8
Miklos Schweitzer 1963_8
Miklos Schweitzer 1963_8
Source:
September 19, 2008
trigonometry
function
integration
search
real analysis
real analysis unsolved
Problem Statement
Let the Fourier series
a
0
2
+
∑
k
≥
1
(
a
k
cos
k
x
+
b
k
sin
k
x
)
\frac{a_0}{2}+ \sum _{k\geq 1}(a_k\cos kx+b_k \sin kx)
2
a
0
+
k
≥
1
∑
(
a
k
cos
k
x
+
b
k
sin
k
x
)
of a function
f
(
x
)
f(x)
f
(
x
)
be absolutely convergent, and let
a
k
2
+
b
k
2
≥
a
k
+
1
2
+
b
k
+
1
2
(
k
=
1
,
2
,
.
.
.
)
.
a^2_k+b^2_k \geq a_{k+1}^2+b_{k+1}^2 \;(k=1,2,...)\ .
a
k
2
+
b
k
2
≥
a
k
+
1
2
+
b
k
+
1
2
(
k
=
1
,
2
,
...
)
.
Show that
1
h
∫
0
2
π
(
f
(
x
+
h
)
−
f
(
x
−
h
)
)
2
d
x
(
h
>
0
)
\frac1h \int_0^{2\pi} (f(x+h)-f(x-h))^2dx \;(h>0)
h
1
∫
0
2
π
(
f
(
x
+
h
)
−
f
(
x
−
h
)
)
2
d
x
(
h
>
0
)
is uniformly bounded in
h
h
h
. [K. Tandori]
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