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
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
was proposed in [3].
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References
J. D’Alembert, Recherches sur la combe que forme une corde tendue mise en vibration I-II. Hist. Acad. Berlin (1747), 2114–249.
Th.M. Rassias and J. Šimša J., Finite Sums Decompositions in Mathematical Analysis. John Wiley & Sons, 1995.
Th.M. Rassias, Problem P286. Aequationes Math. 38 (1989), 11–112.
J. Šimša, The best L 2 -approximation by unite sums of functions with separable variables. Aequationes Math. 43 (1992), 248–263.
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Rassias, T.M. (1997). Problems on finite sums decompositions of functions. In: Bandle, C., Everitt, W.N., Losonczi, L., Walter, W. (eds) General Inequalities 7. ISNM International Series of Numerical Mathematics, vol 123. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8942-1_35
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DOI: https://doi.org/10.1007/978-3-0348-8942-1_35
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