From: Adam Stephanides Subject: Re: sums of periodic real functions Date: Sun, 08 Oct 2000 05:36:38 GMT Newsgroups: sci.math Summary: [missing] Mark Bowron wrote: > > Can the sum of two nonconstant periodic real functions R -> R be > periodic if the quotient of their periods is not rational? Yes. Let w_1, w_2 and w_3 be irrational numbers such that if r*w_1 + s*w_2 + t*w_3 = 0 for integers r, s, t, then r = s = t = 0. If x = r*w_1 + s*w_2 + t*w_3 for some integers r, s, t, define f(x) = s + t and g(x) = r - t; define f(x) = g(x) = 0 otherwise. Then f has period w_1, g has period w_2 and f + g has period w_3. --Adam ============================================================================== From: Adam Stephanides Subject: Re: sums of periodic real functions Date: Sun, 08 Oct 2000 14:05:10 GMT Newsgroups: sci.math denis-feldmann wrote: > > > Mark Bowron wrote: > > > > > > Can the sum of two nonconstant periodic real functions R -> R be > > > periodic if the quotient of their periods is not rational? > Nice counterexample. On the other hand, it is almost obvious that this is > impossible for continuous functions, and i would guess the same for > functions whose discontinuities are not a dense set. The latter seems pretty easy: if f is periodic with period w_1, g with period w_2, f + g with period w_3, w_2/w_1 irrational, then f(x + w_3) - f(x) = g(x) - g(x + w_3). Then f(x + w_3) - f(x) has periods w_1 and w_2, so if it is not discontinuous on a dense set it is constant. Since w_3/w_1 is irrational (otherwise g would have a period with a rational quotient with w_1), f is either constant or discontinuous on a dense set. This proof actually shows that f must be discontinuous on a dense set, provided that g is not doubly periodic with one period rationally related to w_1. > By > the way, functions like f+g are usually called pseudoperiodic (or > quasiperiodic). But i remember having seen many definitions of this (the > usual one is: forall e>0, exists T ,forall x ,|f(x+T)-f(x)| remember really weird ones). Where can i find a summary? I've seen the latter definition referred to as "almost periodic functions." But this definition includes not just sums of two periodic functions, but "Fourier series" of periodic functions with incommensurable periods. This is proved in a very nice book entitled ALMOST PERIODIC FUNCTIONS, iirc (don't remember the author). --Adam ============================================================================== From: Adam Stephanides Subject: Re: sums of periodic real functions Date: Sun, 08 Oct 2000 14:51:00 GMT Newsgroups: sci.math Adam Stephanides wrote: > Then f(x + w_3) - f(x) has periods w_1 and > w_2, so if it is not discontinuous on a dense set it is constant. Since > w_3/w_1 is irrational (otherwise g would have a period with a rational > quotient with w_1), f is either constant or discontinuous on a dense > set. Oops. I just realized I was assuming here that if f(x + w_3) - f(x) is discontinuous on a dense set, so is f. This clearly isn't true in general, even when f is periodic; it may well be true in this particular case, but it's not immediately obvious, and I don't have time to ponder it at the moment. That's what I get for posting in a hurry. --Adam ============================================================================== From: ullrich@math.okstate.edu (David C. Ullrich) Subject: Re: sums of periodic real functions Date: Sun, 08 Oct 2000 14:41:23 GMT Newsgroups: sci.math On Sun, 8 Oct 2000 10:14:08 +0200, "denis-feldmann" wrote: > >Adam Stephanides a écrit dans le message : >39E02FF0.15E8@earthlink.net... >> Mark Bowron wrote: >> > >> > Can the sum of two nonconstant periodic real functions R -> R be >> > periodic if the quotient of their periods is not rational? >> >> Yes. Let w_1, w_2 and w_3 be irrational numbers such that if r*w_1 + >> s*w_2 + t*w_3 = 0 for integers r, s, t, then r = s = t = 0. If x = >> r*w_1 + s*w_2 + t*w_3 for some integers r, s, t, define f(x) = s + t and >> g(x) = r - t; define f(x) = g(x) = 0 otherwise. Then f has period w_1, >> g has period w_2 and f + g has period w_3. > >Nice counterexample. On the other hand, it is almost obvious that this is >impossible for continuous functions, and i would guess the same for >functions whose discontinuities are not a dense set. Is there some simple >description of a larger class of periodic functions for which this (f >a-periodic, g b-periodic, a/b not rational => f+g not periodic) holds? By >the way, functions like f+g are usually called pseudoperiodic (or >quasiperiodic). But i remember having seen many definitions of this (the >usual one is: forall e>0, exists T ,forall x ,|f(x+T)-f(x)|remember really weird ones). Where can i find a summary? There are books on almost periodic functions out there. There was a recent long subthread (I forget exactly where) where I figured out a lot of the stuff I'd learned about them years ago - was trying to see if I actually knew that stuff, so there are various errors there, mostly corrected the next day. At least for continuous functions the answer is no, and yes you could use the theory of AP functions for that: Say f and g have incommensurable periods. Then the spectrum of f is contained in aZ and the spectrum of g is contained in bZ, where a/b is irrational. Now the spectrum of f + g is a subset of aZ union bZ, and if f + g is periodic then the spectrum is contained in a discrete subgroup of R; the only such subgroup contained in aZ union bZ is {0}, so the spectrum of f + g is contained in {0} and hence f + g is constant. (Not that this is the "right" solution(???), but yes AP functions do it, and in fact the stuff proved in that recent thread is enough (there's a proof there that a continuous AP function is determined by its "Fourier coefficients", and this shows what I said here about a function being periodic if and only if its spectrum is contained in a discrete subgroup of R.) The same thing works for periodic functions satisfying an L^2 condition, I believe. >> --Adam > >