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The Hardy-Littlewood maximal function ℳ and the trigonometric function sin x are two central objects in harmonic analysis. We prove that ℳ characterizes sin x in the following way: Let $f \in C^{α} (ℝ, ℝ)$ be a periodic function and α > 1/2. If there exists a real number 0 < γ < ∞ such that the averaging operator $(A_{x} f) (r) = 1 / 2 r \int_{x - r}^{x + r} f (z) d z$ has a critical point at r = γ for every x ∈ ℝ, then f(x) = a + bsin(cx+d) for some a,b,c,d ∈ ℝ. This statement can be used to derive a characterization of trigonometric functions as those nonconstant...

A sharp bound for a sine polynomial

Horst Alzer, Stamatis Koumandos (2003)

Colloquium Mathematicae

We prove that $| \sum_{k = 1}^{n} s i n ((2 k - 1) x) / k | < S i (π) = 1.8519 . . .$ 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).

A special Gaussian rule for trigonometric polynomials.

Milovanovic, Gradimir V., Cvetkovic, Aleksandar S., Stanic, Marija P. (2007)

Banach Journal of Mathematical Analysis [electronic only]

A study of the real Hardy inequality.

Sababheh, Mohamad S. (2009)

JIPAM. Journal of Inequalities in Pure & Applied Mathematics [electronic only]

Adèles et séries trigonométriques spéciales

Yves Meyer (1971/1972)

Séminaire Delange-Pisot-Poitou. Théorie des nombres

An inequality for Chebyshev connection coefficients.

Guyker, James (2006)

JIPAM. Journal of Inequalities in Pure & Applied Mathematics [electronic only]

An inequality for the coefficients of a cosine polynomial

Horst Alzer (1995)

Commentationes Mathematicae Universitatis Carolinae

We prove: If $\frac{1}{2} + \sum_{k = 1}^{n} a_{k} (n) cos (k x) \geq 0 for all x \in [0, 2 π),$ then $1 - a_{k} (n) \geq \frac{1}{2} \frac{k^{2}}{n^{2}} for k = 1, \dots, n .$ The constant $1 / 2$ is the best possible.

An inequality for the maximum of trigonometric polynomials

P. Erdös (1962)

Annales Polonici Mathematici

An inverse Sidon type inequality

S. Fridli (1993)

Studia Mathematica

Sidon proved the inequality named after him in 1939. It is an upper estimate for the integral norm of a linear combination of trigonometric Dirichlet kernels expressed in terms of the coefficients. Since the estimate has many applications for instance in $L^{1}$ convergence problems and summation methods with respect to trigonometric series, newer and newer improvements of the original inequality has been proved by several authors. Most of them are invariant with respect to the rearrangement of the coefficients....

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.

Approximations- und Eindeutigkeitssatz für Exponentialsysteme mit komplexen Exponenten.

Jürgen Elsner (1971)

Journal für die reine und angewandte Mathematik

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