Let a function $f:\mathbb R^2\rightarrow \mathbb R$, such that $f\in C^{1, 2}_b(\mathbb R^2)$ (that is $\partial_x f$, $\partial_y f$ and $\partial_{yy} f$ are continuous and bounded). Is it possible to approximate $f$ and its derivatives by a sequence of functions $$u_n(x, y)=\sum_{i=1}^{k_n}g_n(x)h_n(y),$$ where $g\in C_b^1(\mathbb R)$ and $h\in C^2_b(\mathbb R)$?
2026-03-25 03:02:53.1774407773
Approximation of functions of 2 variables by multiplication of unidimensional functions
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