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A sharp bound for a sine polynomial

Horst Alzer, Stamatis Koumandos (2003)

Colloquium Mathematicae

We prove that | ∑ k = 1 n s i n ( ( 2 k - 1 ) x ) / k | < S i ( π ) = 1 . 8519 . . . for all integers n ≥ 1 and real numbers x. The upper bound Si(π) is best possible. This result refines inequalities due to Fejér (1910) and Lenz (1951).

An inequality for the coefficients of a cosine polynomial

Horst Alzer (1995)

Commentationes Mathematicae Universitatis Carolinae

We prove: If 1 2 + ∑ k = 1 n a k ( n ) cos ( k x ) ≥ 0 for all x ∈ [ 0 , 2 π ) , then 1 - a k ( n ) ≥ 1 2 k 2 n 2 for k = 1 , ⋯ , n . The constant 1 / 2 is the best possible.

An observation on the Turán-Nazarov inequality

Omer Friedland, Yosef Yomdin (2013)

Studia Mathematica

The main observation of this note is that the Lebesgue measure μ in the Turán-Nazarov inequality for exponential polynomials can be replaced with a certain geometric invariant ω ≥ μ, which can be effectively estimated in terms of the metric entropy of a set, and may be nonzero for discrete and even finite sets. While the frequencies (the imaginary parts of the exponents) do not enter the original Turán-Nazarov inequality, they necessarily enter the definition of ω.

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