In the set T_n of trigonometric polynomials f_n of order n with complex coefficients, we consider Weyl (fractional) derivatives f_n^(α) of real nonnegative order α. The inequality ║D_θ^αf_n║_p ≤ B_n(α, θ)_p║f_n║_p for the Weyl-Szegő operator D_θ^αf_n(t) = f_n^α (t)cosθ + f͂_n^α (t) sin θ in the set T_n of trigonometric polynomials is a generalization of Bernstein’s inequality. Such inequalities have been studied for 90 years. G. Szegő obtained the exact inequality ║f′_n cos θ + f͂ ′_n sin θ║_∞ ≤ n║f_n║_∞ in 1928. Later, A. Zygmund (1933) and A.I.Kozko (1998) showed that, for p ≥ 1 and real α ≥ 1, the constant B_n(α, θ)_p equals n^α for all θ ∈ ℝ. The case p = 0 is of additional interest because it is in this case that B_n(α, θ)_p is largest over p ∈ [0, ∞]. In 1994 V. V. Arestov showed that, for θ = π/2 (in the case of the conjugate polynomial) and integer nonnegative α, the quantity B_n(α,π/2)₀ grows exponentially in n as 4^n+o(n). It follows from his result that the behavior of the constant for θ ≠ 2πk is the same. However, in the case θ = 2πo and α ∈ ℕ, Arestov showed in 1979 that the exact constant is n^α. The author investigated Bernstein’s inequality in the case p = 0 for positive noninteger α earlier (2018). The logarithmic asymptotics of the exact constant was obtained: \(\sqrt[n]{{{B_n}{{(\alpha,\,0)}_0}}} \to 4\) as n → ∞. In the present paper, this result is generalized to all θ ∈ ℝ.

中文翻译：

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L 0中三角多项式的Weyl导数的Bernstein–Szegő不等式

在该组Ť _Ñ三角多项式的˚F _Ñ的顺序Ñ具有复系数，我们考虑外尔（分数）衍生物˚F _Ñ^（α）实非负顺序的α。不等式║ d _θ ^α ˚F _Ñ ║ _p ≤乙_Ñ（α，θ）_p ║ ˚F _Ñ ║ _p为外尔-Szegő操作者d _θ ^α ˚F _Ñ（吨）= ˚F _Ñ ^α（吨）COSθ+ ˚F _Ñ ^α（吨）罪θ在设定Ť _Ñ三角多项式的是伯恩斯坦的不等式的概括。这种不平等现象已经研究了90年。G.Szegő获得的精确不等式║ F' _ñ COS θ + F' _Ñ罪θ ║ _∞ ≤ Ñ ║ ˚F _Ñ ║ _∞在1928年以后，A.上Zygmund（1933）和AIKozko（1998）表明，对于p ≥ 1个真实α ≥1，常数乙_ñ（α，θ）_p等于Ñ ^α对于所有θ＆Element;ℝ。的情况下p = 0是额外感兴趣的，因为它是在这种情况下该乙_Ñ（α，θ） _p是最大超过p ∈[0，∞]。在1994年，VV Arestov表明，对于θ =π/ 2（在共轭多项式的情况下）和整数非负α，数量b _n（ α，π/ 2） ₀在n中呈指数增长，为4 ^{n + o（n）}。从他的结果可以看出，常数的行为θ ≠2π ķ是相同的。但是，在的情况下θ =2π ö和α＆Element;ℕ，Arestov显示在1979年，该确切的常数是Ñ ^α。作者研究了早于2018年正非整数α的p = 0情况下的伯恩斯坦不等式。得到精确常数的对数渐近性：\（\ sqrt [n] {{{B_n} {{（\ alpha，\，0）} _ 0}}} \到4 \）为n →∞。在本文件中，这个结果被推广到所有θ＆Element;ℝ。