Abstract
The questions of the characterization of the best approximant for the multivariable functions in the space with mixed integral metric were considered in this article. The criterion of the best approximant in space $$$L_{p_1,...,p_{i-1},1,p_{i+1},...,p_n}$$$ is obtained.
Highlights
We setIs called the best approximation of f in the space Lp1,...,pn by the set Hm. Any polynomial Pm∗ which realizes the ” inf ” on the right-hand side of (1) is said to be a best approximant to f by Hm. In 1973 G.S. Smirnov [2] proved the criterion of the best approximant in the spaces with mixed integral metric for the functions of two variables
Дослiджено питання характеризацiї елемента найкращого наближення для функцiй багатьох змiнних у просторi зi змiшаною iнтегральною метрикою
The questions of the characterization of the best approximant for the multivariable functions in the space with mixed integral metric were considered in this article
Summary
Is called the best approximation of f in the space Lp1,...,pn by the set Hm. Any polynomial Pm∗ which realizes the ” inf ” on the right-hand side of (1) is said to be a best approximant to f by Hm. In 1973 G.S. Smirnov [2] proved the criterion of the best approximant in the spaces with mixed integral metric for the functions of two variables. If at least one pi = 1, these criteroins are true under the condition f − Pm∗ = 0 almost every where on K This restrictionis removed by G.S. Smirnov [3] for the functions of two variables in the spaces Lp,1(I1×I2).
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