A class of positive trigonometric sums. II.
A Class of Positive Trigonometrie Sums.
A majorant problem.
A note on a result of Bateman and Chowla
A note on the best approximation by ridge functions.
A refinement of the Helson-Szegö theorem and the determination of the extremal measures
A rigidity phenomenon for the Hardy-Littlewood maximal function
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 be a periodic function and α > 1/2. If there exists a real number 0 < γ < ∞ such that the averaging operator 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
We prove that 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.
A study of the real Hardy inequality.
Adèles et séries trigonométriques spéciales
An inequality for Chebyshev connection coefficients.
An inequality for the coefficients of a cosine polynomial
We prove: If then The constant is the best possible.
An inequality for the maximum of trigonometric polynomials
An inverse Sidon type inequality
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 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
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.