TY - JOUR
T1 - Roth’s orthogonal function method in discrepancy theory and some new connections
AU - Bilyk, Dmitriy
N1 - Publisher Copyright:
© Springer International Publishing Switzerland 2014.
PY - 2014
Y1 - 2014
N2 - In this survey we give a comprehensive, but gentle introduction to the circle of questions surrounding the classical problems of discrepancy theory, unified by the same approach originated in the work of Klaus Roth (Mathematika 1:73–79, 1954) and based on multiparameter Haar (or other orthogonal) function expansions. Traditionally, the most important estimates of the discrepancy function were obtained using variations of this method. However, despite a large amount of work in this direction, the most important questions in the subject remain wide open, even at the level of conjectures. The area, as well as the method, has enjoyed an outburst of activity due to the recent breakthrough improvement of the higher-dimensional discrepancy bounds and the revealed important connections between this subject and harmonic analysis, probability (small deviation of the Brownian motion), and approximation theory (metric entropy of spaces with mixed smoothness). Without assuming any prior knowledge of the subject, we present the history and different manifestations of the method, its applications to related problems in various fields, and a detailed and intuitive outline of the latest higher-dimensional discrepancy estimate.
AB - In this survey we give a comprehensive, but gentle introduction to the circle of questions surrounding the classical problems of discrepancy theory, unified by the same approach originated in the work of Klaus Roth (Mathematika 1:73–79, 1954) and based on multiparameter Haar (or other orthogonal) function expansions. Traditionally, the most important estimates of the discrepancy function were obtained using variations of this method. However, despite a large amount of work in this direction, the most important questions in the subject remain wide open, even at the level of conjectures. The area, as well as the method, has enjoyed an outburst of activity due to the recent breakthrough improvement of the higher-dimensional discrepancy bounds and the revealed important connections between this subject and harmonic analysis, probability (small deviation of the Brownian motion), and approximation theory (metric entropy of spaces with mixed smoothness). Without assuming any prior knowledge of the subject, we present the history and different manifestations of the method, its applications to related problems in various fields, and a detailed and intuitive outline of the latest higher-dimensional discrepancy estimate.
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U2 - 10.1007/978-3-319-04696-9__2
DO - 10.1007/978-3-319-04696-9__2
M3 - Article
AN - SCOPUS:84921879109
SN - 0075-8434
VL - 2107
SP - 71
EP - 158
JO - Lecture Notes in Mathematics
JF - Lecture Notes in Mathematics
ER -