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The paper "Best $n$-Dimensional approximation to sets of functions" by A. L. Brown in 1964 gave a negative answer to the following question:

Q1: Is there for a given integer $n$ always a best approximation (with respect to the $L_\infty([0,1]^2)$-norm) to a complex valued continuous function of two variables on $[0,1]^2$ by a continuous function of the form $\sum_{j=1}^n f_i(x)g_i(y)$?

Some of the statements at the end of Brown's paper suggest that the following has a positive answer:

Q2: Is there for a given integer $n$ always a best approximation (with respect to the $L_\infty([0,1]^2)$-norm) to a complex valued continuous function of two variables on $[0,1]^2$ by functions of the form $\sum_{j=1}^n f_i(x)g_i(y)$, where $f_i$ and $g_i$ are in $L_\infty([0,1])$?

I cannot find the right reference.

I am also interested to know if the result remains true for functions of two variables in $L_\infty([0,1]^2)$, i.e.,

Q3: Is there for a given integer $n$ always a best approximation (with respect to the $L_\infty([0,1]^2)$-norm) to a function in $L_\infty([0,1]^2)$ by functions of the form $\sum_{j=1}^n f_i(x)g_i(y)$, where $f_i$ and $g_i$ are in $L_\infty([0,1])$?

Thank you in advance.

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  • $\begingroup$ Did you ever find answers to these questions? $\endgroup$
    – Yemon Choi
    Commented May 4, 2015 at 14:09
  • $\begingroup$ I am afraid not. I got around the issue instead. I thought about low rank function approximation for a few weeks, I wrote up my thoughts in Chapter 3 of this math.mit.edu/~ajt/papers/thesis.pdf. But I am very unsatisfied with the overall situation. $\endgroup$
    – alext87
    Commented May 4, 2015 at 14:49

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