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Topic: analysis question on periodic functions
Replies: 7   Last Post: Aug 19, 2013 2:25 PM

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David Bernier

Posts: 3,367
Registered: 12/13/04
analysis question on periodic functions
Posted: Aug 17, 2013 6:21 AM
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Suppose we have an odd continuous function f: R -> R
with period 1 so that f(x+1) = f(x), and f(-x) = -f(x).

Suppose f has mean zero over the unit interval, so
int_{0, 1} f(x) dx = 0, but that
int_{0, 1} | f(x) | dx > 0 (so it's not constantly zero).

Given a real number a> 0, consider the series:

f(a) + f(2a) + f(3a) + ...

with partial sums

S_k (a) = sum_{j = 1 ... k} f(ka).

===


(i) If a is rational, is it necessarily true that the S_k (a) are
bounded in absolute value?


if the answer to (i) were YES, then there's also (ii) below:
if the answer were NO, (ii) might be forgotten.

(ii) If the partial sums S_k (a) are bounded for some
real number a>0, then does it follow that a is a rational
number?



--
abc?



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