All functions can be written as sum of product of $x$ and $y$?

Question

Can all functions of two variables ($x$ and $y$) be written as the sum of the products of a function of $x$ and a function of $y$? E.g.

$a(x,y) = f(x)g(y) + h(x)i(y) + j(x)k(y) ...$

This is a very vague question. Are you allowing infinite series? Must the infinite series converge everywhere? Are you assuming that the function $a$ has some smoothness? As it is, the function $a(x,y) = \begin{cases} 1 \quad (x^2+y^2 < 1) \ 0 \quad (x^2 + y^2 \ge 1) \end{cases}$ might be a counterexample. — Hans Engler, Feb 21 '18 at 23:16
Yes, infinite series would work as it would converge to the value of the original function at the point. — John A., Feb 21 '18 at 23:21
@JohnA. Are you looking for an unique expression for the wjhole domain? — user, Feb 21 '18 at 23:23
Oh whoops, I forgot as I was about to do it after (answer can only be accepted after 10 mins). Thanks for the help! — John A., Feb 23 '18 at 01:50

score 2 · Accepted Answer · answered Feb 21 '18 at 23:11

2

Let consider as counterexample $$f(x,y)=\log (x+y)$$

answered Feb 21 '18 at 23:11

user

Thank you for the counterexample; however, I was thinking could an infinite series do the trick? – John A. Feb 21 '18 at 23:13
That function has a Taylor series about any point in the half plane ${x+y > 0}$, so it is not a counterexample. – Hans Engler Feb 21 '18 at 23:14
@HansEngler but the OP is requesting for an unique espression and not different expressions, am I wrong? – user Feb 21 '18 at 23:16
@gimusi Where does the OP required uniqueness? – Hans Engler Feb 21 '18 at 23:20
@HansEngler It is clear to me from the OP formulation, I ask a clarification on that. – user Feb 21 '18 at 23:22

1 Answers1