On some Extremal Problems of Landau

Révész, Szilárd. "On some Extremal Problems of Landau." Serdica Mathematical Journal 33.1 (2007): 125-162. <http://eudml.org/doc/281520>.

@article{Révész2007,
abstract = {2000 Mathematics Subject Classification: Primary: 42A05. Secondary: 42A82, 11N05.The prime number theorem with error term presents itself as &pi'(x) = ∫2x [dt/ logt] + O ( x e- K logL x). In 1909, Edmund Landau provided a systematic analysis of the proof seeking better values of L and K. At a key point of his 1899 proof de la Vallée Poussin made use of the nonnegative trigonometric polynomial 2/3 (1+cos x)2 = 1+4/3 cosx +1/3 cos2x. Landau considered more general positive definite nonnegative cosine polynomials 1+a1cos x+… + ancos nx ≥ 0, with a1> 1,ak ≥ 0 (k = 1,…,n), and deduced the above error term with L = 1/2 and any K< 1/(2V(a))½, where V(a): = (a1+a2+…+ an)/(( (a1)½-1)2). Thus the extremal problem of finding V: = minV(a) over all admissible coefficients, i.e. polynomials, arises. The question was further studied by Landau and later on by many other eminent mathematicians. The present work surveys these works as well as current questions and ramifications of the theme, starting with a long unnoticed, but rather valuable Bulgarian publication of Lubomir Chakalov.Supported in part in the framework of the Hungarian-French Scientific and Technological Governmental Cooperation, Project # F-10/04. The author was supported in part by the Hungarian National Foundation for Scientific Research, Project #s T-049301, T-049693 and K-61908. This work was accomplished during the author’s stay in Paris under his Marie Curie fellowship, contract # MEIF-CT-2005-022927.},
author = {Révész, Szilárd},
journal = {Serdica Mathematical Journal},
keywords = {Prime Number Formula; Positive Trigonometric Polynomials; Positive Definite Functions; Extremal Problems; Borel Measures; Convexity; Duality; prime number formula; positive trigonometric polynomials; positive definite; functions; extremal problems; Borel measures; convexity; duality},
language = {eng},
number = {1},
pages = {125-162},
publisher = {Institute of Mathematics and Informatics Bulgarian Academy of Sciences},
title = {On some Extremal Problems of Landau},
url = {http://eudml.org/doc/281520},
volume = {33},
year = {2007},
}

TY - JOUR
AU - Révész, Szilárd
TI - On some Extremal Problems of Landau
JO - Serdica Mathematical Journal
PY - 2007
PB - Institute of Mathematics and Informatics Bulgarian Academy of Sciences
VL - 33
IS - 1
SP - 125
EP - 162
AB - 2000 Mathematics Subject Classification: Primary: 42A05. Secondary: 42A82, 11N05.The prime number theorem with error term presents itself as &pi'(x) = ∫2x [dt/ logt] + O ( x e- K logL x). In 1909, Edmund Landau provided a systematic analysis of the proof seeking better values of L and K. At a key point of his 1899 proof de la Vallée Poussin made use of the nonnegative trigonometric polynomial 2/3 (1+cos x)2 = 1+4/3 cosx +1/3 cos2x. Landau considered more general positive definite nonnegative cosine polynomials 1+a1cos x+… + ancos nx ≥ 0, with a1> 1,ak ≥ 0 (k = 1,…,n), and deduced the above error term with L = 1/2 and any K< 1/(2V(a))½, where V(a): = (a1+a2+…+ an)/(( (a1)½-1)2). Thus the extremal problem of finding V: = minV(a) over all admissible coefficients, i.e. polynomials, arises. The question was further studied by Landau and later on by many other eminent mathematicians. The present work surveys these works as well as current questions and ramifications of the theme, starting with a long unnoticed, but rather valuable Bulgarian publication of Lubomir Chakalov.Supported in part in the framework of the Hungarian-French Scientific and Technological Governmental Cooperation, Project # F-10/04. The author was supported in part by the Hungarian National Foundation for Scientific Research, Project #s T-049301, T-049693 and K-61908. This work was accomplished during the author’s stay in Paris under his Marie Curie fellowship, contract # MEIF-CT-2005-022927.
LA - eng
KW - Prime Number Formula; Positive Trigonometric Polynomials; Positive Definite Functions; Extremal Problems; Borel Measures; Convexity; Duality; prime number formula; positive trigonometric polynomials; positive definite; functions; extremal problems; Borel measures; convexity; duality
UR - http://eudml.org/doc/281520
ER -