In our case, the functions $\upsilon$, $g$, and $\eta$ are all defined on the unit circle, so we need a periodic (i.e. circular) convolution. Converting to the Fourier domain therefore involves the inverse DTFT operation, $\mathbf F^\dagger$. Using the convolution theorem, then, we can convert from the ring at critical frequency $\eta$ to the spatial domain on the unit circle $\upsilon$ as follows:
\begin{equation}\begin{aligned} \upsilon &\propto g * \eta = \mathbf F\{ (\mathbf F^\dagger g)(\mathbf F^\dagger \eta) \} \end{aligned}\label{11}\tag{11}\end{equation}
Let's be explicit about this, and give names to these periodic Fourier transforms. Remember, these are now discrete sequences with integer indeces $\alpha\in\mathbb Z$:
\begin{equation}\begin{aligned} \zeta_\alpha &:= \mathbf F^\dagger \eta_\theta \\ \rho_\alpha &:= \mathbf F^\dagger \upsilon_\vartheta. \\ \gamma_\alpha &:= \mathbf F^\dagger g_\theta \end{aligned}\label{12}\tag{12}\end{equation}
I will refer to $\alpha$ as "angular frequent" or "angular mode".
In words, $\zeta_\alpha$ is the circular Fourier transform of the Fourier coefficients at the ring at $\|\boldsymbol\omega\|=1$, $\rho_\alpha$ is the circular Fourier transform of the spatial solution along the unit circle, and $\gamma_\alpha$ is the circular Fourier transform of… something.
* von Mises Aside 2 : What is* $\gamma_\alpha$?
Does $\gamma_\alpha$ reduce to any other familiar objects? Remembering that $g_\theta$ is related to the von Mises distribution, we can actually get an "analytic" expression for $\gamma_\alpha = \mathbf F^\dagger g_\theta$. The Fourier transform of a distribution is called its characteristic function . Using the characteristic function of the von Mises distribution (analytically continued with the shape parameter $\kappa = -i$), we obtain
\begin{equation}\begin{aligned}\gamma_\alpha = 2\pi \operatorname I_\alpha(-i),\end{aligned}\label{13}\tag{13}\end{equation} where $\operatorname I_j$ is the modified Bessel function of the first kind of order $l$.
\begin{equation}\begin{aligned}\operatorname I_\alpha(x) &= i^{-\alpha} \operatorname J_\alpha(ix) = \sum_{m=0}^\infty \frac{1}{m!\, \Gamma(m+\alpha+1)}\left(\frac{x}{2}\right)^{2m+\alpha}\end{aligned}\label{14}\tag{14}\end{equation} This isn't exactly an elementary function, hence why "analytic" was in quotes above. The series definition of the Bessel function is no simpler that saying "The circular Fourier transform of $g_\theta$". Substituting in $\eqref{13}$ in $\eqref{14}$, we get
\begin{equation}\begin{aligned}\gamma_\alpha &= i^{-\alpha} \sum_{m=0}^\infty \frac{(-1)^{m}}{m!\, (m+\alpha)!}\left(\frac{1}{2}\right)^{2m+\alpha}\end{aligned}\label{15}\tag{15}\end{equation} Since $\alpha$ is always an integer, these functions are actually "cylinder functions" (a.k.a. cylindrical harmonics). Perhaps if I were more familiar with special functions and their series definitions, I'd spot something interesting here. For now, it seems to me that the most compact description of $\gamma$ is simply "the circular Fourier transform of $g_\theta = e^{-i c_\theta}$".
With $\zeta$, $\rho$, and $\gamma$, we can write the conversion from spatial-frequency $\zeta$ to spatial-domain $\rho$ as pointwise multiplication:
\begin{equation}\begin{aligned} \rho &\propto \gamma\cdot\zeta \end{aligned}\label{16}\tag{16}\end{equation}
This also implies that we can convert from the spatial domain $\rho$ to the frequency domain $\zeta$ by dividing by $\gamma$, which reminds us that we only need to consider the evolution along the unit circle for solutions of this type:
\begin{equation}\begin{aligned} \zeta &\propto \gamma^{-1} \cdot \rho. \end{aligned}\label{17}\tag{17}\end{equation}
Note that the spatial solution along the unit circle, $\upsilon_\theta$, must be real valued. This means that $\rho_\alpha$ needs to have conjugate symmetry, namely $\rho_{-\alpha} = \rho_\alpha^*$. Inspecting $\eqref{15}$ reveals that the argument (angle in the complex plane) of $\gamma_\alpha$ is determined by $i^{-\alpha}$. The argument of $\gamma_\alpha$ therefore repeats as $\{1,-i,-1,i\}$ for $\alpha % 4 \in {0,1,2,3}$. The spatial Fourier components $\eta_\theta$ does have an interesting symmetry inherited from the conjugate symmetry of a 2D Fourier transform of a real-valued function: $\eta_{\theta+\pi} = \eta_\theta^*$. So, the real component of $\eta$ is $\pi$ periodic, and affects only even angular modes $\alpha$, and the imaginary component of $\eta$ affects only odd angular modes.
What did we learn?
We learned that 2D solutions consisting entirely of plane waves at a single frequency $\omega_c$ ("monochromatic") can be described entirely by a 1D ring of spatial frequency components $\eta_\theta$. Amusingly, this implies that we can also describe these solutions using only their values along the unit circle in the spatial domain. The 2D pattern is defined by the dynamics along a 1D boundary.
To review, we considered 2D solutions $u_{\boldsymbol x}$, and their Fourier transform $\xi_{\boldsymbol\omega}$. We converted from $\xi_{\boldsymbol\omega}$ to $u_{\boldsymbol x}$ using the inverse 2D Fourier transform. If $u_{\boldsymbol x}$ is "monochromatic", i.e. consist entirely of waves at a single frequency (which we will take to be $1$), we only need to consider frequency components $\eta_{\theta} = \xi(\mathbf c_\theta, \mathbf s_\theta)$ along the ring at this frequency. The frequency components are related to the spatial solution evaluated on the unit circle as $\upsilon_\vartheta \propto\int_{d\theta}\eta_{\theta} e^{i \cos(\theta - \vartheta)}$. This is a 1D circular convolution with a function $g_\theta = e^{-i c_\theta}$, which is connected to the von Mises distribution and cylindrical harmonics.
Applying the convolution theorem, we see that the spatial-domain angular-frequency components are related to the frequency-domain angular-frequency components by a simple scalar multiplication with a fixed function we called $\gamma$. The low-order angular frequency components may provide a compact low-order approximation of the solution.
The 1D unitary DTFT
\begin{equation}\begin{aligned} \mathbf F \zeta &= \frac {1}{\sqrt{2\pi}} \sum_{\alpha\in\mathbb Z} \zeta_\alpha e^{-i\theta\alpha} \\ \mathbf F^\dagger \eta &= \frac {1}{\sqrt{2\pi}} \int_{d\theta} \eta_\theta e^{i\theta\alpha} \end{aligned}\label{f1}\end{equation}
The 2D unitary FT
\begin{equation}\begin{aligned} \mathcal F u &= \frac 1 {2\pi} \iint_{d\boldsymbol x} u_{\boldsymbol x} e^{-i\langle \boldsymbol\omega, \boldsymbol x\rangle} \\ \mathcal F^\dagger \xi &= \frac 1 {2\pi} \iint_{d\boldsymbol\omega} \xi_{\boldsymbol\omega} e^{i\langle \boldsymbol\omega, \boldsymbol x\rangle} \end{aligned}\label{f2}\end{equation}
The 2D FT truncated to a 1D ring at unit frequency (double check normalization here, this might be wrong)
\begin{equation}\begin{aligned} \mathfrak F u &= \frac 1 {2\pi} \iint_{d\boldsymbol x} u_{\boldsymbol x} e^{-i\langle \boldsymbol\omega_\theta, \boldsymbol x\rangle} \\ \mathfrak F^\dagger \eta &= \frac 1 {\sqrt{2\pi}} \int_{d\theta} \eta_{\theta} e^{i \langle \boldsymbol\omega_\theta, \boldsymbol x\rangle } \\ \boldsymbol\omega_\theta &:= \{\mathbf c_\theta, \mathbf s_\theta\} \end{aligned}\label{f3}\end{equation}
The 1D transformation relating the spatial domain on the unit circle to the ring at unit frequency for monochromatic solutions:
\begin{equation}\begin{aligned} \mathcal G \upsilon &= \frac 1 {\sqrt{2\pi}} \int_{d\vartheta} \upsilon_{\vartheta} e^{-i \cos(\theta - \vartheta)} \\ \mathcal G^\dagger \eta &= \frac 1 {\sqrt{2\pi}} \int_{d\theta} \eta_{\theta} e^{i \cos(\theta - \vartheta)}. \end{aligned}\label{f4}\end{equation}
For the most part, this is wandering in the wilderness, but there is one hint of helpful intuition. In many cases, Turing patterns take on some interesting symmetries, for example, forming stripes (1 wave), or checker-boards (2 waves), or spots with the translation symmetry of a hexagonal/triangular lattice (3 waves), or perhaps even quasicrystals (4 or more waves). In these cases, rotational symmetry is broken. Instead of a ring at $\omega_c$, we might get solutions with some angular rotational symmetry. In this case, the circular Fourier transform, of the Fourier transform of the spatial solutions along the ring at critical frequency, may tell us all we need to know. This hints that perhaps we only need the low-order components, perhaps $\alpha \in \{0,\pm 1,\pm 2,\pm 3\}$, to capture most patterns (and maybe a handful more if we want to capture quasicrystalline solutions).
I am tempted to return to $\eqref{1}$ and explore how we might use this low order "angular mode" description to analyze the dynamics. The procedure looks something like this:
First, we consider a spatially homogeneous steady-state solution $u_0$:
\begin{equation}\begin{aligned} u_0 &= f( k_0 u_0 ), \end{aligned}\label{q1}\end{equation}
where $k_0$ is the DC (zero frequency, mean) term in the convolution $\mathbf K*$. We then consider the evolution of solutions $u_{t, \boldsymbol x} = \epsilon_{t, \boldsymbol x} + u_0$ that are an $\epsilon_{t, \boldsymbol x}$ perturbation away from this homogeneous steady-state. We usually then consider the low-order terms in a Taylor expansion of the nonliterary $f(\cdot)$:
\begin{equation}\begin{aligned} \dot \epsilon &= \dot u - \dot u_0 = \dot u \\&= -u + f( [\mathbf K * u]_{t, \boldsymbol x} ) \\&= -u+ \sum_{n=0}^\infty \frac 1 {n!} f^{(n)}(u_0) (\mathbf K * u)^n \\&= -\epsilon+ \sum_{n=1}^\infty a_n \cdot (\mathbf K * \epsilon)^n. \end{aligned}\label{q2}\end{equation}
where we have abbreviated the coefficients of the Taylor expansion $f^{(n)}(u_0)/n!$ as $a_n$.
First, relate the spatial domain $\epsilon$ to the spatial frequency domain $\xi = \mathcal F \epsilon$. Define $k_{\boldsymbol\omega}$ as the Fourier coefficients of the convolution $\mathbf K *$. We are faced with the awkward situation that the exponentiation $(\cdot)^n$ is diagonal in the spatial basis, but convolution $\mathbf K *$ is diagonal in the spatial frequency basis.
We can therefore either transform to the spatial domain to exponentiate, or use the somewhat unconventional iterated convolution $(\cdot)^{*n}$ notation in the frequency domain, reflecting the fact that integer powers of $n$ in the spatial domain become iterated convolutions in the frequency domain:
\begin{equation}\begin{aligned} \dot \xi &= -\xi+ \sum_{n=1}^\infty a_n \mathcal F\{\mathcal F^\dagger\{k \xi\}^n \} \\ &= -\xi+ \sum_{n=1}^\infty a_n [k \xi]^{*n}. \end{aligned}\label{q3}\end{equation}
Now, maybe it's OK to assume that $\epsilon$ is a "monochromatic" solution with an especially simply low-order description. In this case, we care only about the components $\eta_\theta = \xi(\mathbf c_\theta, \mathbf s_\theta)$. Did I mention the kernel $\mathbf K$ is isotropic, i.e. the same in all directions? Let's assume that it is. In this case, the only component of $k$ that we care about is $k_c$, the Fourier coefficient of $\mathbf K *$ at the critical frequency. This is the same regardless of the direction $\theta$. We can write this reduced-order representation of $\eqref{q3}$ as:
\begin{equation}\begin{aligned} \dot \eta &\propto -\eta + \sum_{n=1}^\infty a_n \mathfrak F\{\mathfrak F^\dagger \{k_c \eta\}^n \}, \end{aligned}\label{q4}\end{equation}
where we've switched the notation for the Fourier transform to $\mathfrak F$ to remind us that our "spatial frequency domain" is now the 1D object $\eta$, as opposed to the full 2D $\xi$.
We need to be careful. So far, our equations $\eqref{q2}$ and $\eqref{q3}$ really do describe the full nonlinear dynamics of $u$, in the spatial and spatial-frequency domains respectively (assuming the Taylor series converges). We haven't truncated the series expansion yet. However, when we restrict are solutions to those that can be described as a sum of plane waves at fixed frequency, we are modifying the dynamics. In general, the nonlinearity will generate some frequencies outside $\omega_c$, so this low-order approximation is not closed. Dropping these these terms may change things, and it remains to be seen whether this simplified system can capture the dynamics we care about. In particular, restricting to $\eta_\theta$ from $\xi_{\boldsymbol\omega}$ is tantamount to assuming that the only nonzero Fourier coefficients of $\mathbf K$ are the DC term and the term at $\omega_c$.
Now, we could go a step further and change coordinates to $\zeta = \mathbf F^\dagger \eta$ and again to $\rho=\gamma\zeta$, but this will get messy. Let's try to make our lives easier. Our equations are defined by the following:
- A convolution, which is diagonal in the spatial frequency domain.
- A pointwise nonlinearity in the spatial domain, which we've expanded as a power series
If we restrict ourselves to unit frequency, step (1) is just scalar multiplication by a single Fourier component $k_c$. This commutes with any integral transform (these are linear). The only "hard" step is the pointwise nonlinearity. For this, we should convert to the spatial domain (and back again). If we work with $\rho$, this is just the circular Fourier transform of the spatial domain, so we can convert to/from the spatial domain using the DTFT. The dynamics on $\rho$ can therefore be written as:
\begin{equation}\begin{aligned} \dot \rho &\propto -\rho+ \sum_{n=1}^\infty a_n \mathbf F^\dagger\{[\mathbf F\{k_c \rho \}]^n\} \\ &= -\rho+ \sum_{n=1}^\infty a_n \{k_c\rho\}^{*n} \\ &= -\rho + \sum_{n=1}^\infty a_n k_c^n \rho^{*n}. \end{aligned}\label{q6}\end{equation}
This looks worryingly simple, and may indicate that something has gone wrong (we've thrown away too much by assuming a single spatial frequency). We can also convert back to $\upsilon_\vartheta = \mathbf F \rho$, the spatial solution on the unit circle, and find:
\begin{equation}\begin{aligned} \dot \upsilon_\vartheta &\propto -\upsilon_\vartheta + \sum_{n=1}^\infty a_n (k_c \upsilon_\vartheta)^n \\ & = -\upsilon_\vartheta+ f[ k_c \upsilon_\vartheta ] - \upsilon_0. \end{aligned}\label{q7}\end{equation}
This is strange, and obviously wrong: We have lost all spatial coupling! Is the mistake conceptual (this 1D ring parameterization is no good), or did I make a calculation error?
