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On Sine Series with Monotone Coefficients
Author(s) -
Hartman Philip,
Wintner Aurel
Publication year - 1953
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/s1-28.1.102
Subject(s) - citation , monotone polygon , series (stratigraphy) , library science , mathematics , computer science , information retrieval , history , biology , paleontology , geometry
summary:In this paper we obtain estimates of the sum of double sine series near the origin, with monotone coefficients tending to zero. In particular (if the coefficients $a_{k,l}$ satisfy certain conditions) the following order equality is proved $$ g(x,y)\sim mna_{m,n}+\frac mn\sum _{l=1}^{n-1}la_{m,l}+\frac nm\sum _{k=1}^{m-1}ka_{k,n}+\frac 1{mn}\sum _{l=1}^{n-1}\sum _{k=1}^{m-1}kla_{k,l}, $$ where $x\in (\frac {\pi }{m+1}, \frac {\pi }m]$, $ y\in (\frac {\pi }{n+1}, \frac {\pi }n]$, $ m, n=1,2,\dots $