Wikipedia:Reference desk/Archives/Mathematics/2019 May 25

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May 25

Functions from convolutions

For idle curiosity....if I have some unknown functions f,g,h but I know their convolutions f*g, g*h, and f*h, is this enough information to figure out what f,g,h are? If not, what additional information is needed to pin them down? --HappyCamper 12:53, 25 May 2019 (UTC)

After Fourier transform, the convolutions become products, so you get 3 algebraic equations with 3 unknowns. Count Iblis (talk) 14:21, 25 May 2019 (UTC)

Periods and subgroups of the real numbers

Given any function f from the real numbers R to itself, a period of f is any real number p for which f(x + p) = f(x) for all real numbers x. Clearly, the periods of any function (including zero) always form a subgroup of the additive group of real numbers. For which subgroups G of (R, +) does there exist a continuous function f for which the subgroup of periods of f is exactly equal to G? Note that if continuity is not required, then such a function always exists: just let f be the indicator function of G. GeoffreyT2000 (talk) 14:24, 25 May 2019 (UTC)

For subgroup {0, 2π, 4π, 6π, ...} there exists such a function: $\sin x$. Ruslik_Zero 20:57, 25 May 2019 (UTC)

In fact, thanks to this ProofWiki result, the answers are exactly just R itself and its cyclic subgroups. Indeed, the subgroup of periods of a continuous function f must be a closed subgroup because it is the intersection $\bigcap _{x\in \mathbb {R} }f_{x}^{-1}(f_{x}(0))$, where, for any real number x, f_x is the function sending a real number y to f(x + y). If it is a proper subgroup, then it cannot be dense, and so must be cyclic per the linked ProofWiki page. Clearly, any aperiodic function will work for the trivial subgroup containing just zero, while any constant function will work for the improper subgroup. For the cyclic subgroup aZ, where a is a nonzero real number, either $f(x)=\sin({\frac {2\pi {x}}{a}})$ or $f(x)=\cos({\frac {2\pi {x}}{a}})$ will work. GeoffreyT2000 (talk) 03:46, 27 May 2019 (UTC)

And you may like to consider the analogous question for a continuous function on Rⁿ. The set of p in Rⁿ for which f(x + p) = f(x) for all x in Rⁿ is a closed subgroup of Rⁿ, and any closed subgroup of Rⁿ is the set of periods of some continuous function. In fact, a closed subgroup of Rⁿ is the direct sum of a discrete group (isomorphic to some Z^k) and a linear subspace (isomorphic to some R^h). pma 21:54, 28 May 2019 (UTC)