We study a new phenomenon of the behaviour of widths with respect to the optimality of trigonometric system. It is shown that the trigonometric system is optimal in the sense of Kolmogorov widths in the case of “super-high” and “super-small” smoothness but is not optimal in the intermediate cases. Bernstein’s widths behave differently when compared with Kolmogorov in the case of “super-small” smoothness. However, in the case of “super-high” smoothness Kolmogorov and Bernstein widths behave similarly, i.e. are realized by trigonometric polynomials.

中文翻译：

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关于三角系统的最优性

我们研究了关于三角系统最优性的宽度行为的新现象。结果表明，在“超高”和“超小”光滑度的情况下，三角系统在Kolmogorov宽度意义上是最佳的，但在中间情况下不是最佳的。在“超小”平滑度的情况下，与Kolmogorov相比，Bernstein的宽度行为有所不同。但是，在“超高”平滑度的情况下，Kolmogorov和Bernstein宽度的行为类似，即通过三角多项式实现。