Zentralblatt MATH

Publications of (and about) Paul Erdös
Zbl.No:  106.27701
Autor:  Erdös, Pál
Title:  About an estimation problem of Zahorski (In English)
Source:  Colloq. Math. 7, 167-170 (1960).
Review:  The author refutes Z.Zahorski's conjecture [Problem 168, Colloquium Math. 4, 241 (1957)] concerning the estimate of the following integral:

int02\pi |\cos n1 x+\cos n2 x+···+\cos nk x| dx = O(log nk).

He proves that
I: There exists a sequence nk for which int02 \pi | sum1k \cos ni x | dx > Ck 1/2 -\epsilon.
II: There exists a sequence n, for which int02\pi | sum1k \cos ni x | dx = \sqrt\pi \sqrt{nk}+o(\sqrt {nk}), which proves that O(\sqrt {nk}) is the best estimate.
Reviewer:  Y.M.Chen
Classif.:  * 26A06 One-variable calculus
                   41A17 Inequalities in approximation
Index Words:  approximation and series expansion

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