## Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  106.27702
Autor:  Erdös, Pál
Title:  An inequality for the maximum of trigonometric polynomials (In English)
Source:  Ann. Pol. Math. 12, 151-154 (1962).
Review:  Let

fn(\theta) = sumk = 1n (ak \cos k\theta+bk \sin k \theta)

be a trigonometric polynomial with real coefficients. Put M = max0 \leq \theta < 2a |fn (\theta)|. The author conjectures that there exists an absolute constant c > 0 such that

(1)   M \geq {1+c \over \sqrt 2} \left{sumk = 1n (ak2+bk2) \right} ½,

with c \leq \sqrt 2-1.
Theorem: Assume that max1 \leq k \leq n (max |ak|,|bk|) = 1 and that sumk = 1n (ak2+bk2) = An. Then there exists a c = cA > 0 depending on A for which limA ––> 0 cA = 0 and \TagsOnLeft

M > {1+cA \over \sqrt 2} \left{sumk = 1n (ak2+bk2) \right} ½.    (2)

Reviewer:  Y.M.Chen
Classif.:  * 42A05 Trigonometric polynomials
Index Words:  approximation and series expansion

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