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Is there any function$f:\Bbb R^2\to\Bbb R$ which has no representation of the form below?

$$f(x,y)=\sum_{n=1}^{\infty} g_n(x)h_n(y)\quad(x,y\in \Bbb R).$$Editor's note: A possible source of motivation for this question lies in a trick used to find solutions to linear partial diferential equations, in particular Laplace's equation in two dimensions. As a first step, a solution of the form $f(x,y)=g(x)h(y)$ is sought; and then more-general solutions of the form displayed above can be generated. While it is far from obvious that all solutions to the 2-D Laplace equation must be of this form, it is also hard to find a counterexample, namely a solution that cannot be thus structured. (Indeed, is such a solution known at all?)

If we relax the conditions so as not to require even continuity of $f$ (let alone its being a solution of Laplace's equation), then there is in principle more room to find a counterexample. Thus it seems that the question posed above may be answerable. In fact, while this editor is unable to come up with a counterexample, his (admittedly unreliable) intuition says that there probably is one and that the answer is yes.

John Bentin
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Zeinab
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  • Try $e^{xy}$.${}{}$ – Seewoo Lee Dec 15 '18 at 08:22
  • @JohnBentin It may be all of those things, but it still lacks context. – jgon Dec 15 '18 at 16:15
  • @jgon: Could you supply an example of context that would be needed to allow this question to be posted? – John Bentin Dec 15 '18 at 16:19
  • @JohnBentin I can't give the context, since I'm not the OP, but as examples, I would accept: The OP's motivation for asking this question, e.g. Where is it from? Did the OP come up with it themselves? If so, how? Why do they think it is an interesting question? Other forms of context I would accept are what they OP's thoughts about the question are. Do they think it's true or false? If so why or why not? – jgon Dec 15 '18 at 16:22
  • It's really not hard to add context to a question if you're asking it, since if you're asking you must have a reason to ask and some thoughts on the question, or you wouldn't be bothering. – jgon Dec 15 '18 at 16:23
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    The problem is related to (but not solved by) the Kolmogorov-Arnold representation theorem. I agree if the OP is interested in the problem, then they should add context beyond mere problem statement to their Question. I would also accept context added by @JohnBentin if so motivated. – hardmath Dec 15 '18 at 16:37
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    @jgon: For me, the question is incomparably more interesting than any details of how or why the OP came up with the question. Also, I can't see how such details would help in answering the question. – John Bentin Dec 15 '18 at 16:46
  • @JohnBentin In that case you can either edit your own context to this question (in your own voice) or ask this question as a new question and supply your own context by explaining why you think it is interesting.

    Don't get me wrong, I'm guilty of answering PSQs that I think are interesting as much as anyone, but that doesn't change the community policy.

     – jgon Dec 15 '18 at 17:06
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    @jgon: In principle, this question could be stolen, dressed up with some plausible-sounding "context", and re-posted, probably to gain a good response and some upvotes. But I think that the OP should get the credit. I simply hope that enough people who dig what a lovely question it is come along and vote to restore it. – John Bentin Dec 15 '18 at 17:54
  • @JohnBentin, So edit in your own context then. I gave you two suggestions for courses of action that would result in a version of this question with context I would consider sufficient. As it is now, it's not a good question. – jgon Dec 15 '18 at 17:58
  • @jgon: I have taken your advice and added some motivation. Would this be enough to win your vote to reopen? – John Bentin Dec 15 '18 at 18:49
  • @JohnBentin Yes, I personally think that's a good context, voted. – jgon Dec 15 '18 at 18:50
  • @hardmath: Is the context that I have added enough to win your vote to reopen? – John Bentin Dec 15 '18 at 19:28
  • @hardmath: If the question is of general interest, and actually posted, then many users of this site will appreciate it. Is it so important, then, whether any answer is over the head of one user (the OP), or whether that user really understood what he was posing? – John Bentin Dec 15 '18 at 20:16
  • @JohnBentin: While it is important to know whether the answers address what the OP "really wanted to learn", and in the short term this takes precedence over what future users may take from the Question, I have voted to reopen. The OP has not weighed in on your RFC/edit, but they did visit the site since you revised the Question, and did not rollback your edit. I'm willing to take silence as consent in this circumstance. – hardmath Dec 16 '18 at 15:12
  • Reposted to MathOverflow: https://mathoverflow.net/questions/320019/is-this-a-known-question-about-the-expression-of-a-function-on-bbb-r2-as-an – Nate Eldredge Jan 04 '19 at 15:56
  • @JohnBentin: This question is not stolen at all. I have never seen such a question before. My major is harmonic analysis. Especially, I work on BSE algebras. When I was trying to find a counterexample for a claim, I need to find a two variable function witch has no sigma representation as it is asked. – Zeinab Mar 26 '19 at 13:51

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