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Real Analysis
book

Real Analysis

by V. Karunakaran
May 2024
Intermediate to advanced content levelIntermediate to advanced
585 pages
15h 38m
English
Pearson India
Content preview from Real Analysis
“real: chapter_11” 2011/5/23 0:43 page 11 #11
Fourier Series 11-11
M-test). An application of Theorem 11.2.14 now gives that the Fourier
series of f converges to f uniformly on [−π, π].
Theorem 11.2.16 Define the Dirichlet kernel D
n
(t) by
D
n
(t) =
1
2
+
n
k=1
cos kt =
sin
n+
1
2
t
2 sin(t/2)
for t = 0
n +
1
2
for t = 0.
Then D
n
(t) T
n
and D
n
(t) satisfies the following properties.
(i) For f C
2π
,s
n
(f )(x) =
1
π
π
π
f (x + t)D
n
(t)dt.
(ii) D
n
(t) = D
n
(t)(i.e. D
n
(t) is even).
(iii)
1
π
π
π
D
n
(t)dt =
2
π
π
0
D
n
(t)dt = 1.
(iv) |D
n
(t)|≤n +
1
2
and D
n
(0) = n +
1
2
.
(v)
|sin
n+
1
2
t|
t
≤|D
n
(t)|≤
π
2t
for 0 < t .
(vi) If λ
n
=
1
π
π
π
|D
n
(t)|dt, then
4
π
2
log n λ
n
3 + log n.
Proof We first ...
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Publisher Resources

ISBN: 9781299447561Publisher Website