Can any function $f(x,y)$ of 2 variables be written as $$f(x,y)=\sum_{n=1}^\infty g_n(x) h_n(y)$$
If not, what is the condition that allows it to be written as such?
PS: I've read this thread before.
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what is the difference to the linked question? – miracle173 Sep 09 '17 at 05:51
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@miracle173 I know that it can't be written as single product of $g(x)h(y)$ but can it be written in series of them? – mon Sep 09 '17 at 05:54
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what does "written in series" mean? "g(x)h(y)" is a product. – miracle173 Sep 09 '17 at 05:58
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@miracle173 $Sigma g_n(x) h_n(y)$ – mon Sep 09 '17 at 05:59
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If the function can be expanded in a power series, then the power series is the series you are looking for. – Zach Boyd Sep 09 '17 at 06:01
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then you should formulate your question accordingly. – miracle173 Sep 09 '17 at 06:02
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1It's true for a large collection of functions, but only in a certain mode of convergence for the series. See for example this question. – shalop Sep 09 '17 at 06:19