Search All of the Math Forum:

Views expressed in these public forums are not endorsed by NCTM or The Math Forum.

Notice: We are no longer accepting new posts, but the forums will continue to be readable.

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

 Messages: [ Previous | Next ]
 David Bernier Posts: 3,892 Registered: 12/13/04
analysis question on periodic functions
Posted: Aug 17, 2013 6:21 AM

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?

Date Subject Author
8/17/13 David Bernier
8/17/13 David Bernier
8/17/13 David C. Ullrich
8/17/13 David Bernier
8/17/13 quasi
8/18/13 David C. Ullrich
8/18/13 quasi
8/19/13 quasi