The characterization of the best approximant for the multivariable functions in the space $L_{p_1,..,p_n;\Omega }$

The questions of the characterization of the best approximant in spaces of multivariable functions with mixed integral metric with weight were considered in this article. The general form of a bounded linear functional and the criterion of the best approximant in these spaces are obtained.

Let Ω(x) = Ω(x 1 , .., x n ) be the nonnegative summable on If Ω(x) = 1, ∀x ∈ K, then L p;Ω = L p = L p 1 ,...,pn . We set If Ω(x) = 1, ∀x ∈ K, we will write |f | p k ,...,p i . Consider also the classes L p 1 ,...,pn (where at least one p i = ∞) of functions f respectively with norms f ∞,p 2 ,...,pn;Ω = ess sup In 1973 G.S. Smirnov [1] proved the criterion of the best approximant in the spaces with mixed integral metric for the functions of two variables. V.M. Traktynska [3] extended this result to the multivariable functions in the spaces L p 1 ,p 2 ,...,pn . V.M. Traktynska and M.E. Tkachenko [4] proved the criterion of the best approximant in the spaces with mixed integral metric with weight for the functions of two variables. The purpose of this article is getting the general form of a bounded linear functional and the criterion of best approximant in the space L p 1 ,...,pn;Ω .
Applying Gelder's inequality and Fubini's Theorem, we obtain for We can get similar inequality in the case when some p i = 1 except p 1 . So for 1 ≤ p i , q i < ∞, 1 ≤ i ≤ n, p 1 = 1, we get the inequality: In the case when p 1 = 1, 1 < p i < ∞, 1 < i ≤ n, we will have: ess sup Similarly inequality holds true in the case when some p i = 1.
where sup distributed to functions g ∈ L q;Ω such us g q;Ω = 1 and g ⊥ H m , that iś sup on the right-hand side of (3) is realized on functions g ∈ L q;Ω with the norm g q;Ω = 1.
In particular, if f ∈ L p;Ω then where sup on the right-hand side of (4) distributed to all functions g ∈ L q;Ω , g q;Ω ≤ 1.
Lemma 1. If f p;Ω > 0 then sup on the right-hand side of (4) distributed to the function The function g 0 will be unique (if at least one of p i = 1, assuming that f (x) = 0 almost everywhere on K).
Theorem 3. The polynomial P * m ∈ H m is the best approximant for f ∈ L p;Ω \H m if and only if there exists a function g 0 ∈ L q;Ω that satisfies the conditions: Theorem 3 is the implementation of the general criterion for the best approximant.
Theorem 4 is completely proved.