Finding Functional form for a given Scaling Condition - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T00:29:15Z http://mathoverflow.net/feeds/question/19186 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/19186/finding-functional-form-for-a-given-scaling-condition Finding Functional form for a given Scaling Condition jonalm 2010-03-24T10:47:24Z 2010-03-26T06:06:53Z <p>Dear all</p> <p>While studying the overlap distribution for two random Cantor sets (long story made short), I came across the following problem.</p> <p>$G(k)$ is a complex valued function, and satisfy the following condition:</p> <p>$G(k\mu) = G(k)^2+ \beta$</p> <p>with $\beta,\mu$ constant (in my case $\beta=\frac{2}{9}, \mu = \frac{4}{3}$)</p> <p>Is there a way to find the functional form of $G(k)$ which satisfy the condition?</p> <p>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?</p> <p>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:</p> <p><a href="http://dl.dropbox.com/u/483049/xy.pdf" rel="nofollow">http://dl.dropbox.com/u/483049/xy.pdf</a></p> <p>-- jon</p> http://mathoverflow.net/questions/19186/finding-functional-form-for-a-given-scaling-condition/19196#19196 Answer by Julián Aguirre for Finding Functional form for a given Scaling Condition Julián Aguirre 2010-03-24T14:47:06Z 2010-03-24T14:47:06Z <p>You do not give any smoothness requirement; I will look for an analytic $G$: $$G(k)=\sum_{n=0}^\infty a_nk^n.$$ In what follows, I assume also that $\mu=4/3$ and $\beta=2/9$. Expanding in a power series both sides of the equation and equating coefficients, we get that $a_0=1/3$ or $a_0=2/3$. In the first case we obtain the constant solution $G(k)=1/3$. But in the second case, we find a one parameter family of (formal) solutions, parametrized by the value of $a_1$: $$a_0=\frac23,\quad a_1\in\mathbb{C},\quad a_n=\frac{1}{\mu^n-\mu}\sum_{i=1}^{n-1}a_ia_{n-i},\quad n>2.$$ For $a_1=0$ we obtain the constant solution $G(k)=2/3$. For other values of $a_1$, one should check that the series has a positive radius of convergence.</p> <p>Another way of obtainig solutions is the following. Choose an arbitrary function $h\colon[1,\mu]\to\mathbb{C}$, and define $G(k)=h(k)$ if $1\le k&lt;\mu$; for $\mu\le k&lt;\mu^2$, let $G(k)=G(k/\mu)^2+\beta$; iterate this procedure to define $G$ on $[1,\infty)$. Now, for $1/\mu\le k&lt;1$, let $G(k)=\pm\sqrt{G(\mu k)^2-\beta}$; iterate the procedure to define $G$ on $(0,1)$. Conditions can be impposed on the arbitrary function $h$ to make $G$ continuous, for instance ($G(\mu)=G(1)^2+\beta$).</p>