Additively separable function

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Definition

For a function of two variables

Suppose F is a function of two variables x and y. We say that F is additively separable if there exist functions f,g of one variable such that:

F(x,y) = f(x) + g(y)

on the entire domain of F.

Note that the concept of additively separable is sensitive to the coordinate system, i.e., if we change the coordinate system, a function that was originally additively separable need not remain additively separable.

For a function of many variables

Suppose F is a function of n variables x_1,x_2,\dots,x_n. We say that F is completely additively separable if there exist functions f_1,f_2,\dots,f_n, each a function of one variable, such that:

F(x_1,x_2,\dots,x_n) = f_1(x_1) + f_2(x_2) + \dots + f_n(x_n)

(note that the subscripts here are not to be confused with subscripts used for partial derivatives).

There is a weaker notion of partially additively separable: if we express the set \{ 1,2,\dots,n\} as a union of two disjoint subsets A,B, F is additively separable with respect to the partition if there exist functions f_A,f_B such that:

F(x_1,x_2,\dots,x_n) = f_A(\mbox{only the variables } x_i, i \in A) + f_B(\mbox{only the variables } x_i, i \in B)

Partial derivatives

Additively separable functions are the exceptions to the general rule that value of partial derivative depends on all inputs.

Version type Statement about first-order partial derivatives Statement about second-order mixed partial derivatives
additively separable function F of two variables x,y, both pieces are differentiable functions, written as F(x,y) = f(x) + g(y) F_x(x,y) = f'(x) (independent of y)
F_y(x,y) = g'(y) (independent of x)
F_{xy}(x,y) = 0
F_{yx}(x,y) = 0
completely additively separable function F of n variables x_1,x_2,\dots,x_n, written as f_1(x_1) + \dots + f_n(x_n) F_{x_i}(x_1,x_2,\dots,x_n) = f_i'(x_i) for each i. Note that each first-order partial depends only on that variable and not on the others. F_{x_ix_j}(x_1,x_2,\dots,x_n) = 0 for each i,j.
partially additively separable function F(x_1,x_2,\dots,x_n) equals f_A(\mbox{only the variables } x_i, i \in A) + f_B(\mbox{only the variables } x_i, i \in B) Each first-order partial of F with respect to a variable in A equals the corresponding first-order partial of f_A, and in particular depends only on the variables within A.
Each first-order partial of F with respect to a variable in B equals the corresponding first-order partial of f_B, and in particular depends only on the variables within B.
Any second-order mixed partial involving a variable in A and a variable in B is zero.