Periodic Functions Problem

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Here’s a neat little problem that I learned about during a party in my first year of graduate school. I don’t know where it’s from originally, but I got it from Joe Miller:

Show that there exist two periodic functions f,g:\mathbb{R}\rightarrow\mathbb{R} whose sum is the identity function:

f(x)+g(x)=x for all x\in\mathbb{R}.

Here periodic has the standard definition: a function f:\mathbb{R}\rightarrow\mathbb{R} is periodic if there exists a constant c>0 such that f(x+c)=f(x) for x\in\mathbb{R}.

Obviously the functions f and g can’t be continuous, since any continuous periodic function is bounded. Indeed, it is possible to show that f and g can’t both be measurable.

In case you’re interested, this paper gives a complete criterion for determining whether a function can be written as a sum of periodic functions with specified periods, and this paper investigates the question of whether such functions can be measurable.

15 Responses to “Periodic Functions Problem”

  1. Omar Says:

    Am I missing something or is this pretty easy?

    Just take a basis for the reals over the rationals, and define f and g on the basis so that their sum is the identity for ever basis element. Be sure to set f equal to zero on at least one basis element a, and g equal to zero on some other basis element b. Extend f and g by linearity (over the rationals). Now, f+g is the identity, f has period a and g has period b.

    Please don’t ask me for a solution that doesn’t use the axiom of choice…

  2. Jim Belk Says:

    Yeah that’s about it. I’m not sure, but I doubt that there’s a solution that doesn’t involve the axiom of choice. It’s not that hard of a problem, but I’ve always found the statement really surprising . . . two periodic functions summing to give the identity.

    Also, the solution gives a neat way of thinking about periodic functions. Note that you can choose for f and g to each be zero for uncountably many different basis elements, in which case both functions have uncountably many different periods.

    • Peter Says:

      Jim: as you suspect, any solution must involve the axiom of choice (or some similar principle beyond just ZF set theory). As pointed out in the post, any solution here involves a non-measurable function, while Solovay (iirc) showed that in ZF, it is consistent that all functions from R to R are measurable.

      • Peter Luthy Says:

        Yeah, Solovay is probably the right guy. I know he’s responsible for showing that one needs more than just ZF to construct a non-measurable set.

  3. John Armstrong Says:

    Not just different periods, incommensurable periods. That is, no multiple of one period of the function is a multiple of another period. Weird. Eerie.

  4. A note on the Periodic Functions Problem « The Unapologetic Mathematician Says:

    […] note on the Periodic Functions Problem Over at The Everything Seminar, Jim Belk mentions an interesting little problem. Show that there exist two periodic functions […]

  5. David Radcliffe Says:

    Some extensions: (1) show that every polynomial function is a finite sum of periodic functions, and (2) show that the exponential function is not a finite sum of periodic functions.

  6. Jim Belk Says:

    Interesting . . . for the polynomial, you can start by expressing x as the sum of n different periodic functions:

    x = f_1 + f_2 + … + f_n

    You have to arrange it so that any n-1 of these functions have a common period. (This is easily accomplished: for each choice of n-1 functions, choose a basis element for which they are all zero.)

    Since any n-1 of these functions have a common period, any monomial in f_1,…,f_n of degree strictly less than n will be periodic. It follows that any polynomial in x of degree less than n is the sum of periodic functions. In fact, if you collect terms with the same period, you can arrange for any polynomial of degree n-1 to be the sum of at most n periodic functions.

  7. Ewan Delanoy Says:

    A related question : what if we demand that f and g be continuous but relax the range and only ask for f and g to be defined on a finite interval [0,M] ? (the peridocity condition is then redefined in the obvious way, f(x+T)=f(x) whenever x and x+T are both in the range). To fix ideas, say we want f to be 1-periodic and g to be T-periodic, where 0<T<1. When T is rational there will be a largest M such that a solution exists on [0,M]. When T is irrational I think that there should be solutions for any M.

  8. munadiakrom Says:

    To define a periodic function is not difficult. Could you give sample how to define a periodic polynomial function!
    Thanks.

  9. prof dr mircea orasanu Says:

    must to mention that definition of PERIODICA FUNCTIONS have another form observed prof dr mircea orasanu and prof drd horia orasanu and followed that these can be extended ,then can be extended as double periodical functions specially complex Analytical functions and can be applied to complete integral ELLIPTICAL

  10. prof dr mircea orasanu Says:

    and as is observed that periodic functions are as example trigonometric functions and double periodic functions are zeta functions, complete elliptical integral of first kind or second kind , or three kind elliptical integrals considered prof dr mircea orasanu and prof drd horia orasanu and followed that for these are with ìmportance properties

  11. prof dr mircea orasanu Says:

    for evidence of periodic functions or double periodic functions as in complex are observed by prof dr mircea orasanu and prof drd horia orasanu is followed that really honest about it, none of the manipulations we do with Lagrangians in optimization are decidedly intuitive. Mechanistically, Lagrangians give powerful methods to derive algorithms, understand sensitivities to assumptions, and generate lower bounds. But the functionals themselves always just seem to pop out of thin air. Why are Lagrangian methods so effective in optimization, even when the associated problems are nonconvex?
    In this post, I’m going to “derive” Lagrangians in two very different ways: one by pattern matching against the implicit function theorem and one via penalty functions.

  12. prof dr mircea orasanu Says:

    these are most important aspects pf these problem observed prof dr mircea orasanu and prof drd horia orasanu and followed that must developed

  13. prof dr mircea orasanu Says:

    most periodical functions appear for functions simple periodic s and multiple periodical functions observed prof dr mircea orasanu and prof drd horia orasanu and observed that these can be used in LAGRANGIAN OPTIMIZATIONS and CONSTRAINTS

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