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Given the following renormalization-group-like recurrence equation for the function $p_k(x)$

\begin{equation} p_{k+1}(x) = \int dy \, p_k(y) \, \delta(f(y)-x), \; k\geq0, \; \; \; \; (1) \end{equation}

where $x$ is a real variable, and $f(x)$ a given function, I want to find the fixed point of this equation, i.e., a function $p_{\ast}$ such that $p_{\ast}(x) = \int dy \, p_{\ast}(y) \, \delta(f(y)-x)$.

Please note that Eq. (1) here is just an example that I am using to ask my question clearly, and that I have no pretentions of rigor here.

I solve Eq. (1) numerically by representing $p_{k}$ in a discrete form, namely as a set of $N$ numbers $\pi^k_1, \cdots, \pi^k_N$, which represent the values of $p_k(x)$ at $N$ points $x_1, \cdots, x_N$ on a grid. Given $\pi^k_1, \cdots, \pi^k_N$, I compute $\pi^{k+1}_1, \cdots, \pi^{k+1}_N$ from Eq. (1), iterate $K$ times for $k=0,1, ...., K-1$, and obtain an estimate the fixed point $\pi^K_1, \cdots, \pi^K_N$.

This estimate depends on two numbers: $N$ and $K$. The 'exact' fixed point is obtained by letting both these numbers go to infinity.

Are you aware of a proper method, or some references, on how to do this two-parameter extrapolation?

Thank you.

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  • $\begingroup$ Might this question be better suited for Mathematics? $\endgroup$
    – ACuriousMind
    Feb 17, 2017 at 12:47

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