Decompose a function of multiple variables to product of functions of single variable

Asked Jul 12 '24 at 09:23

Active Jul 12 '24 at 09:23

Viewed 130 times

For simplicity, we consider a two variable function $f = f(x,y): \mathbb{R}^2 \to \mathbb{R}$. As stated in this post, in general, we can't write $$f(x,y) = g(x) h(y)$$ as a product of single variable functions, for some $g = g(x)$ and $h = h(y)$.

My question is following: Can we decompose $f$ into an infinite sum of functions of separation variable, that is, do there exist sequence $(g_n)_{n=1}^{\infty}$ and $(h_n)_{n=1}^{\infty}$ of functions of single variable such that $$f(x,y) = \sum_{n=1}^{\infty} g_n(x) h_n(y)$$ for all $(x,y) \in \mathbb{R}^2$?

Any hint or example is welcome.

asked Jul 12 '24 at 09:23

nyan

Sufficiently nice functions can be written in this form: just look at the Taylor series. – Kavi Rama Murthy Jul 12 '24 at 09:31
What sense of convergence are you looking for? – strugglingmathematician Jul 12 '24 at 09:35
@strugglingmathematician For simplicity, we consider in the sense of pointwise convergence. But, my goal is convergence in $L^2$... – nyan Jul 12 '24 at 09:50
@nyan For the case of $L^2$ (or more generally $L^p$) see here. – jd27 Jul 12 '24 at 16:36
$L^2$ is (sometimes) far more simple than pointwise convergence. Consider Carleson's theorem re convergence of Fourier series. – strugglingmathematician Jul 13 '24 at 06:51

Decompose a function of multiple variables to product of functions of single variable

0 Answers0