RE: Determining whether a function is periodic
- To: mathgroup at smc.vnet.net
- Subject: [mg35953] RE: [mg35932] Determining whether a function is periodic
- From: "DrBob" <majort at cox-internet.com>
- Date: Fri, 9 Aug 2002 05:18:01 -0400 (EDT)
- Reply-to: <drbob at bigfoot.com>
- Sender: owner-wri-mathgroup at wolfram.com
I don't think you've made yourself clear.
First of all, you defined a functional ff, but then said you're LOOKING
FOR the functional. If it's already defined, you don't have to LOOK for
it. I'm guessing that f is unknown, so you're actually looking for f.
"My problem is finding the value(s) for x" doesn't make that clear.
Secondly, your ff as defined isn't a functional, it's a function. It
takes a number u to a number ff[u], not functions to numbers. Anyway,
the period of a functional would be a function, not a number. (Any
function in the null space of a linear functional would be a "period" of
the functional.)
Third, your ff has period p if f has 0 integral over every interval of
length k p that starts at u (or, equivalently, every interval of length
p that starts at u + k p). Sin[x] has 0 integral over every interval of
length k*2Pi that starts at 0, for instance. Hence there are infinitely
many f that make ff periodic.
To see that, just write down what the following actually means:
ff[u] = ff[u + k*p] where k=integer, p=period
which is
0 == ff[u + k*p] - ff[u] == Integrate[f[x],{x,u,k p}]
Bobby
-----Original Message-----
From: Deirdre Stewart [mailto:dstewart at hebel.net]
To: mathgroup at smc.vnet.net
Subject: [mg35953] [mg35932] Determining whether a function is periodic
Hi,
I've got the feeling I'm overlooking something obvious...
In the framework of an optimization problem, I'm tinkering with a class
of
real integral functionals - i.e. functions defined by an integral - of
a real
variable.
I define a functional ff(u) as the integral of a function f(x) with
integration boundaries [0,u]
In Mathematica notation, ff[u] := Integrate[ f(x), {x,0,u} ]
I know that the functional I'm looking for is periodic for some values
of u.
My problem is finding the value(s) for x such that ff(u) is perfectly
periodic.
The functional ff is quite well-behaved, and uniformly converges towards
a periodic function in u's vicinity.
The human eye is quite good at detecting periods in plots, but surely
there's
a clever way to make Mathematica automatically assess a function's
periodicity, given the period value p one expects ?
ff(u) = ff(u + k*p) where k=integer, p=period
As the period is known, I could e.g. look at the magnitude of the
expected
Dirac deltas after calculating a Fourier Transform of the integral
functional,
but this seems quite inefficient and prone to numerical errors.
Any suggestions as to how to detect periodicity in Mathematica would be
most welcome...
Deirdre Stewart