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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 17 #17
Fourier Series 11-17
1
π
|t|≤δ
|f (x + t) f (x)|k
n
(t)dt
+
1
π
δ≤|t|≤π
|f (x + t) f (x)|k
n
(t)dt
1
π
π
π
k
n
(t)dt +
2
π
||f ||
δ≤|t|≤π
k
n
(t)dt.
Using the properties of the summability kernel, it follows that the
right-hand side of the above inequality can be made as small as we
want provided n is large. Further this estimate is independent of x. This
completes the proof.
Corollary 11.2.22 (Fejer’s Theorem) Let f C
2π
and s
n
(f ),
the n
th
partial sum of the Fourier series of f . Then
n
f )(x) =
1
n
(
s
0
(f )(x) + s
1
(f )(x) +···+s
n1
(f )(x)
)
converges uniformly to
f (x) on [−π , π] as n →∞.
Proof By (i) of Theorem ...
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Publisher Resources

ISBN: 9781299447561Publisher Website