## Zentralblatt MATH

Publications of (and about) Paul Erdös

Zbl.No:  023.02201
Autor:  Erdös, Paul; Turán, Paul
Title:  On the uniformly-dense distribution of certain sequences of points. (In English)
Source:  Ann. of Math., II. Ser. 41, 162-173 (1940).
Review:  Le théorème suivant est démontré: soient \phi1(n), \phi2(n),...,\phin(n), n = 1,2,3,..., des nombres satisfaisant aux conditions 0 \leq \phi1(n) < \phi2(n) < ··· < \phin(n) \leq \pi. Si les polynômes \omegan(z) = prod\nu = 1n (z-\cos \phi\nu(n)) satisfont à l'inégalité |\omegan(z)| \leq 2-n A(n), où A(n) est une fonction croissante tendant vers l'infini, alors pour chaque intervalle (\alpha,\beta), 0 \leq \alpha < \beta \leq \pi on a

|{sum\nu}\alpha \leq \phi\nu(n) \leq \beta {sum\nu} 1-\frac{\beta-\alpha}{\pi} n | < {8 \over log 3} (n log A(n))^ 1/2 .

The following theorem is proved: Let \phi1(n), \phi2(n),...,\phin(n), n = 1,2,3,..., be numbers satisfying the conditions 0 \leq \phi1(n) < \phi2(n) < ··· < \phin(n) \leq \pi. If the polynomials \omegan(z) = prod\nu = 1n (z-\cos \phi\nu(n)) satisfy the inequality |\omegan(z)| \leq 2-n A(n), where A(n) is an increasing function tending to infinity, then for each interval (\alpha,\beta), 0 \leq \alpha < \beta \leq \pi the following inequality is valid

|{sum\nu}\alpha \leq \phi\nu(n) \leq \beta {sum\nu} 1-\frac{\beta-\alpha}{\pi} n | < {8 \over log 3} (n log A(n))^ 1/2 .

Reviewer:  N.Obrechkoff (Sofia)
Classif.:  * 42A05 Trigonometric polynomials
Index Words:  Approximation of functions, orthogonal series developments

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