Re: Sum of periodic functions with incommensurate periods


David C. Ullrich wrote:
> On 14 Oct 2005 09:14:00 -0700, "David M Einstein" <Deinst@xxxxxxxxx>
> wrote:
>
> >David C. Ullrich wrote:
> >> On Thu, 13 Oct 2005 09:12:43 EDT, Victor <victor_88@xxxxxxx> wrote:
> >>
> >> >Greetings!
> >> >
> >> >I'm desperately looking for a proof (or a disproof) of the following fact.
> >> >
> >> >Let f(x) and g(x) be two periodic functions R->R with least positive periods T1 and T2 respectively, and T1 and T2 are incommensurate (say, T1=1, T2=sqrt(2) ). Then f(x)+g(x) is not periodic.
> >> >
> >> >There are absolutely no restrictions on f(x) and g(x), they are not necessarily continious or have any other properties, they are just functions from R to R.
> >> >
> >> >Is is true? Is yes, then how can it be proven?
> >> >
> >> >Thank you very much in advance!
> >>
> >> I mentioned this to the irritatingly smart guy upstairs. A few hours
> >> later he told me he'd given a counterexample - I found this
> >> sufficiently irritating that I came up with my own counterexample
> >> a few hours later.
> >>
> >> The counterexample is not so hard, surprised nobody's posted an
> >> example yet. Our two examples are based on the same idea, although
> >> his was more elegant - I'll give my version.
> >>
> >> First, you should note that a periodic function need not have
> >> a smallest positive period! For example the characteristic function
> >> of the rationals is periodic... it's not clear to me whether
> >> "with least positive periods T1 and T2 respectively" was
> >> supposed to be an additional hypothesis or just a declaration
> >> of a name for these things that need not exist.
> >>
> >> It's easy to see that there exist periodic f and g with no
> >> common period, such that f + g is periodic (which may or
> >> may not answer your question). Note that R is a vector
> >> space over Q. Although people don't usually think of it
> >> that way, a "period" of a linear operator is precisely
> >> a non-zero element of the kernel. If you look at bases
> >> it's easy to "construct" examples of Q-linear f and g
> >> with non-trivial kernels, with kernels disjoint except
> >> for 0, such that f + g also has a non-trivial kernel,
> >> and that gives a counterexample, _except_ that these
> >> f and g do not _have_ smallest positive periods. You
> >> can get an example where f and g have smallest positive
> >> periods by a similar construction, with functions
> >> that are not quite Q-linear:
> >>
> >> Let B be a basis for R as a Q-vector space,
> >> and say b1, b2, b3 are distinct elements of B.
> >> "b" will always denote an element of B. Say
> >> c_b(x) is the coefficient of b in when x is
> >> written as a Q-linear combination of elements of B.
> >>
> >> Let
> >>
> >> f(sum_b c_b(x) b) = b1 + sum_{b <> b1} c_b(x) b,
> >>
> >> unless c_b1(x) is an integer, in which case
> >>
> >> f(sum_b c_b(x) b) = sum_{b <> b1} c_b(x) b.
> >>
> >> Then f has period b1, and the zero set of f
> >> is precisely the integer multiples of b1, so
> >> f has no smaller positive period.
> >>
> >> Let b' = b unless b = b3; let (b3)' = -b3.
> >> Let
> >>
> >> g(sum_b c_b(x) b) = b2 + sum_{b <> b2} c_b(x) b',
> >>
> >> unless c_b2(x) is an integer, in which case
> >>
> >> g(sum_b c_b(x) b) = sum_{b <> b2} c_b(x) b'.
> >>
> >> Then g has smallest positive period b2, and
> >> f+g has period b3.
> >>
> >> I believe that's right - I can fill in details
> >> for any of the assertions that are not clear,
> >> I think.
> >>
> >> In that last f+g has arbitrarily small positive
> >> periods. Probably that's more interesting than
> >> getting f+g to have a smallest positive period;
> >> if you'd prefer that f+g also have a smallest
> >> positive period then you could probably modify
> >> the above, jiggling the way f and g depend on
> >> c_b3(x).
> >
> >I do not see how arbitrarily small periods occur.
> >If we perform a parallel construction in
> >Q(sqrt(2),sqrt(3)), with b1=1, b2=sqr(2),
> >b3=sqrt(3), f+g has minimal period sqrt(3).
> >Am I missing something?
>
> I don't know exactly what you mean by "parallel
> construction", unless it's the same construction...
>
> Anyway, in the example above any rational multiple
> of b3 is a period of f+g. This is because f+g
> is depends on all the c_b _except c_b3: if
> y = x + r b3 where r is rational then c_b(y)
> = c_b(x) for b <> b3, and hence (f+g)(y) = (f+g)(x).

Ugh. There I go again confusing the rationals with the integers.
>
> You should pay no attention to this construction
> anyway - it's been vastly simplified and improved
> (see parallel post).
>
> >> I don't know whether f and g can be measurable
> >> (They can't be tempered distributions...)
> >>
> >> ************************
> >>
> >> David C. Ullrich
> >
>
>
> David C. Ullrich

.

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