User jonalm - MathOverflow

Fourier transform of Analytic Functions

2010-05-06T08:57:50Z

Forgive me if this question does not meet the bar for this forum. But i would really appreciated some help.

I'm trying to construct a function according to some conditions in the frequency domain of the Fourier transformation. I want the function to be analytic and real when I transform it back to the time domain. The Fourier transformation of $f$ has of course some symmetry criteria to make $f$ real. But what about the Analytic property. As an analytic function imply some convergent power series expansion, and the Fourier transform of a polynomial is a sum of derivatives of Delta functions, I assume that there is a corresponding criteria of the Fourier transformation.

So the question is: If a function $f:\mathbb{R}\rightarrow \mathbb{R}$ is assumed to be analytic, what is the corresponding criteria for the Fourier transform of the function $\mathcal{F}[f] (k)$?

Edit: what I am trying to construct is probability distribution with the following condition

$f(x/\mu)/\mu=\frac{2}{3} f(x) + \frac{1}{3} (f\ast f)(x)\quad$ where $\ast$ mark the convolution, and $\mu=\frac{4}{3}$. $f$ is positive and real for $x\in [0,\infty)$

Taking the fourier transformation make the condition simpler:

$\tilde f(\mu k) = \frac{2}{3}\tilde f(k) + \frac{1}{3}\tilde f^2(k)$

So my problem is to construct $f$ (I am in particular interested in the tail behavior) and I try to use the properties of $\tilde f$. I posted a similar problem a while ago (see here). Julián Aguirre answered how to construct $\tilde f$ if it is analytic. But the inverse transformation of the power expansion is an infinite sum of derivatives of Delta functions, and is of little help.

Number of config. of a binary string invariant under cyclic permutation.

2010-05-30T12:43:44Z

The following combinatorial problem has bothered me quite a bit. I guess people smarter than me have given the problem some taught as the problem has obvious applications (e.g. to the Ising model), but I have not found any solution on the web (this might be because I don't know the proper terminology).

Anyways, here is the problem:

Consider a string of $N$ binary variables, $\uparrow$ and $\downarrow$. The string will have $2^N$ different configurations. Now impose a symmetry to the system; two configurations are equal if you can get from one to the other by cyclic permutation or by reversal of the string (or a combination of these two symmetries). How many unique configurations will the string have?

For 1 $\uparrow$ and $N-1$ $\downarrow$ there will only be 1 unique configuration. For 2 $\uparrow$ and $N-2$ $\downarrow$ there will be $N/2$ configurations if $N$ is even and $(N-1)/2$ configurations if $N$ is odd. But if you take 3 $\uparrow$ and $N-3$ $\downarrow$, it is no longer clear (at least not to me), how one efficiently should count the number of possible configurations.

I would really appreciated some help, or references on relevant literature.

Problem with a Long Range Correlated Time Series

2010-04-05T19:28:57Z

Consider a stochastic process $X_t$ , $t \in 1,2,3,..,N $.

$X_t$ is a Bernoulli variable and $\Pr(X_t=1) = p$ for all $t$. The Autocovariance function $\gamma(|s-t|)= E[(X_t - p)(X_s -p)]$ is given

$ \gamma(k) = \frac{1}{2} (|k-1|^{2H} - 2|k|^{2H} + |k+1|^{2H}). $

For a constant $H\in (0,1)$ This is the same autocovariance as for fractional gaussian noise (increments of the fractional brownian motion), and give a autocovariance which falls like a power law when $k$ goes to infinity.

Let X and Y be process with the given properties, I am interested in the following probability distribution:

$ \Pr\left(\sum_{i=0}^N X_i Y_i = k\right) $

That is the distribution of the overlap of two such processes. For $H=1/2$ the process is not correlated and I have the simple result that $\Pr(X_t Y_t)=p^2$, and that

$ \Pr\left(\sum_{i=0}^N X_i Y_i = k\right) = {N \choose k} p^{2k} (1-p^2)^{N-k}. $

But for $H\neq 1/2$, I do not know how to deal with the long range correlation. Is there a way to proceed on this problem? I regret i never took a class in Stochastic Analysis, and I really hope the question makes sense. Any help or input would be highly appreciated.

Finding Functional form for a given Scaling Condition

2010-03-24T10:47:24Z

Dear all

While studying the overlap distribution for two random Cantor sets (long story made short), I came across the following problem.

$G(k)$ is a complex valued function, and satisfy the following condition:

$G(k\mu) = G(k)^2+ \beta$

with $\beta,\mu$ constant (in my case $\beta=\frac{2}{9}, \mu = \frac{4}{3}$)

Is there a way to find the functional form of $G(k)$ which satisfy the condition?

Note that for $\beta = 0$, $G(k)=\exp\left(a k^{\log_\mu 2}\right)$, ($a$ konstant) will satisfy the condition (easily verified), but I have no idea on how to find a solution for non-zero $\beta$. I'm a not a math student (I'm studying physics), but I have never seen problems like this before. Is there a way to find analytical expression for $G(k)$? Possible as an expansion?

I can generate a function which has this property on the computer. Writing $G(k)= x(k) + i y(k)$, with $x(k)=x(-k)$ and $y(k)=-y(-k)$ the function should look something like this:

http://dl.dropbox.com/u/483049/xy.pdf

-- jon

Comment by jonalm

2010-06-01T09:33:15Z

Thank you for the input. @Streve: do you know of a reference for the solution of the 1D Ising model related to Necklace combinatorics.

Comment by jonalm

2010-05-06T19:24:58Z

Thanks for the input. But I do need the linearity of the transformation (see the edit of my question), and that is not the case according to wikipedia.

Comment by jonalm

2010-05-06T19:17:32Z

Thanks for your interest fedja, I edited my question and explained what I'm trying to construct.

Comment by jonalm

2010-05-06T10:16:02Z

But I guess there exist functions which are not Schwartz, but has a well defined Fourier transform (?). In either case, is there a general way to express a Schwartz function. Like a series expansion?

Comment by jonalm

2010-04-07T04:26:49Z

What exactly do I need to specify? I think any process will do, as long as it satisfy the autocorrelation and the symmetry.

Comment by jonalm

2010-04-06T18:52:46Z

I also have a symmetry condition. But I'm not sure how to state it properly. All path have a non-zero probability, and if a path have some probability $f(p)$, possibly depending on the marginal probability $p$, then the probability of the 'mirror' path (1->0 and 0->1), should have the probability $f(1−p)$. Will this condition suffice for the question to be well posed?

Comment by jonalm

2010-04-06T12:26:43Z

Thank you for the input. I can generate a sequence like this by writing the N binary variates as $X_n=\mathbb{I}_{0}({Z_n})$, where $\mathbb{I}_{A}(b)=0$ if $b\in A$ and 1 else, and where $Z_n$ is a sum of a set of $N(N+1)/2$ Poisson variables. The algorithm is explained in detail in: "A simple method for Generating Correlated Binary Variates" (jstor.org.stable/2684925). Writing the whole expression in terms of Poisson variables is very messy, so I was hoping to utilize the structure of the covariance function directly to calculate the distribution I'm interested in.

Comment by jonalm

2010-03-27T13:44:55Z

Thank you so much Julián. But this solution raise another question for me. You see that G(k)= $\tilde F(k)$ is the Fourier transform of F(x). And I want find an expression of F(x), but every term in the inverse Fourier transformation (from the series expansion) seem to diverge. Is it a way to tackle this problem? Assume $a_1= const.i$ as this will make F(x) a real function.

Comment by jonalm

2010-03-24T13:30:37Z

Got it. Thanks.

Comment by jonalm

2010-03-24T13:20:25Z

@Harald: I use Firefox 3.6.2 on Mac. It show like this: dl.dropbox.com/u/483049/screenshot.tiff

Comment by jonalm

2010-03-24T12:03:57Z

The latex notation does not seem to show properly in Firefox. I don't know the reason. Hope the question is clear.