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On the non-equivalence of rearranged Walsh and trigonometric systems in L p

Aicke Hinrichs, Jörg Wenzel (2003)

Studia Mathematica

We consider the question of whether the trigonometric system can be equivalent to some rearrangement of the Walsh system in L p for some p ≠ 2. We show that this question is closely related to a combinatorial problem. This enables us to prove non-equivalence for a number of rearrangements. Previously this was known for the Walsh-Paley order only.

On the pointwise strong approximation by (C,1)(E,1) means

Włodzimierz Łenski, Bogdan Roszak (2008)

Banach Center Publications

We present an estimate of the (C,1)(E,1)-strong means with mixed powers of the Fourier series of a function f ∈ L 2 π as a generalization of the result obtained by M. Yildrim and F. Karakus. Some corollaries on the norm approximation are also given.

On the proof of Erdős' inequality

Lai-Yi Zhu, Da-Peng Zhou (2017)

Czechoslovak Mathematical Journal

Using undergraduate calculus, we give a direct elementary proof of a sharp Markov-type inequality ∥ p ' ∥ [ - 1 , 1 ] ≤ 1 2 ∥ p ∥ [ - 1 , 1 ] for a constrained polynomial p of degree at most n , initially claimed by P. Erdős, which is different from the one in the paper of T. Erdélyi (2015). Whereafter, we give the situations on which the equality holds. On the basis of this inequality, we study the monotone polynomial which has only real zeros all but one outside of the interval ( - 1 , 1 ) and establish a new asymptotically sharp inequality.

On the quantitative Fatou property

A. Kamaly, A. M. Stokolos (2002)

Colloquium Mathematicae

The result of this article together with [1] and [4] gives a full quantitative description of a Fatou type property for functions from Hardy classes in the upper half plane.

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