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I am not a mathematician, so what I ask might be trivial, however I couldn't find something relevant in the web. My question is the following:

Is there a formal notation for functions that comply the following

$$f(x_1, ..., x_n) = g(f(x_1), ..., g(f(x_n))$$

and if there is such a property, how is it called? The current, intuitive name I can think of is decomposable functions, but I don't know if formally there is such a thing.

Example 1: An additive function would be decomposable.

Example 2: A multiplicative function would also be decomposable

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3 Answers 3

I don't know of any standard terminology for this property. I think "decomposability" is a fine term when defined in context. An interesting example of such a function is $f(f(x)) = x^2+1$.

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I think the notion of homomorphism in universal algebra might be what you are looking for.

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Multivariate functions representable as a combination of univariate functions arise in several places in mathematics. For instance, ordinary differential equations $y'(x) = f(x,y)$ where $f(x,y)=g(x)h(y)$ are called equations with separable variables. Such equations can be easily solved by integrating $G(x)=\int g(x)dx$ and $F(y)=\int\frac{dy}{h(y)}$ separately, and solving the equation $F(y)=G(x)$ for $y$.

Also a method of separation of variables for solving homogeneous partial differential equations in two variables is based on searching for a solution in the form of a product of univariate functions. For instance for Laplace equation $\Delta u = 0$ considering a solution of the form $u(x,y)=g(x)h(y)$ leads to a pair of ordinary differential equations and ultimately to a method of finding a solution to boundary value problems for Laplace equation in the form of a series $u(x,y) = \Sigma_{n=1}^\infty B_n g_n(x)h_n(y)$ where $g_n(x)$ and $h_n(y)$ are the functions corresponding to the $n$-th eigenvalue of the boundary value problem.

A special case where a multivariate function $f(x,y_1,...,y_k)$ is a linear combination of the form $\Sigma_{i=1}^n g_i(x)h_i(y_1,...,y_k)$, is of special interest. Such functions had been called decomposable in the literature (cf. M. Čadek, J. Šimša, Decomposable functions of several variables, Aequationes Mathematicae, 1990, and references thereof).

For sufficiently differentiable functions there is an easy test of decomposability. Consider a function of two variables $f(x,y)$. Define its Wronskian as

$\mathcal{W}_n=\mathop{\textrm{det}}\left[\frac{\partial^{i+j}f}{\partial x^i\partial y^j}\right].$

Note that this differs from the usual definition of Wronskian of a sequence of univariate functions. This is can be seen as a usual Wronskian with respect to the first variable of the sequence of partial derivatives in the second variable.

We can say that a sufficiently differentiable function $f(x,y)$ is decomposable as $f(x,y)=\Sigma_{i=1}^n g_i(x)h_i(y)$ for some $n$ if and only if $\mathcal{W}_n = 0$ and $\mathcal{W}_{n-1} \neq 0$.

Getting back to the original question of naming, the word decomposable is alright and has been used in the literature, however variants on the theme of separation of variables or separability might fit in some contexts.

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