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Bernstein–Szegő Inequality for the Weyl Derivative of Trigonometric Polynomials in L 0
Proceedings of the Steklov Institute of Mathematics ( IF 0.4 ) Pub Date : 2020-05-28 , DOI: 10.1134/s0081543820020108
A. O. Leont’eva

In the set Tn of trigonometric polynomials fn of order n with complex coefficients, we consider Weyl (fractional) derivatives fn(α) of real nonnegative order α. The inequality ║DθαfnpBn(α, θ)pfnp for the Weyl-Szegő operator Dθαfn(t) = fnα (t)cosθ + nα (t) sin θ in the set Tn 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 θnfn in 1928. Later, A. Zygmund (1933) and A.I.Kozko (1998) showed that, for p ≥ 1 and real α ≥ 1, the constant Bn(α, θ)p equals nα for all θ ∈ ℝ. The case p = 0 is of additional interest because it is in this case that Bn(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 Bn(α,π/2)0 grows exponentially in n as 4n+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 nα. 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:

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 ÑpÑ(α,θ)p˚F Ñp为外尔-Szegő操作者d θ α ˚F Ñ)= ˚F Ñ α)COSθ+ ˚F Ñ α)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' ÑθÑ˚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ñ(α,θ)p等于Ñ α对于所有θ∈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) 0n中呈指数增长,为4 n + on。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 öα∈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:\(\ sqrt [n] {{{B_n} {{(\ alpha,\,0)} _ 0}}} \到4 \)n →∞。在本文件中,这个结果被推广到所有θ∈ℝ。
更新日期:2020-05-28
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