Problems on finite sums decompositions of functions

Abstract

Let f: X×Y → ℝ be function and an integer n ≥ 1 where X, Y are arbitrary nonempty sets. Find the best approxmation $f\left( {x,\,y} \right) \approx \sum\limits_{{i = 1}}^n {{f_i}(x){g_i}(y)}$ with respect to the supremum norm ∥f∥ = sup {∣f∣(x,y)∣: x ∈ X and y ∈ Y}. This is even an open problem for polynomial functions f defined on the Cartesian product of two real intervals. If one changes the norm then the problem seem to be totally open. It will also be interested to study the best approximation problem

$$ f\left({{x_1}, \ldots, {x_k}} \right) \approx \sum\limits_{{{i_1} = 1}}^{{{m_1}}} \cdots \sum\limits_{{{i_k} = 1}}^{{{m_k}}} {{\alpha_{{{i_1} \ldots {i_k}}}}f_{{{i_1}}}^1\left({{x_1}} \right) \cdots f_{{{i_k}}}^k\left({{x_k}} \right)} $$

for functions f in more than two variables, even if the L ²-norm is considered. The corresponding problem of the best L ²-approximation (for functions of two variables only) has been solved in Chapter 7 of the book by Th.M. Rassias and J. Šimša [2] (see also [4]). The problem of finding a necessary and sufficient condition for a function f in several variables to be represented by

$$ f\left({{x_1},{x_2}, \ldots, {x_k}} \right) = \sum\limits_{{i = 1}}^n {{f_{{{i_1}}}}\left( {{x_1}} \right){f_{{{i_2}}}}\left({{x_2}} \right) \cdots {f_{{{i_k}}}}\left({{x_k}} \right)} $$

was proposed in [3].