Geodesic
On the behavior near the origin of double sine series with monotone coefficients
Mathematica Bohemica, Tome 134 (2009) no. 3, pp. 255-273

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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 $.
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 $.
DOI : 10.21136/MB.2009.140660
Classification : 42A16, 42A20
Keywords: double sine series; sum of a double sine series with monotone coefficients 
Krasniqi, Xhevat Z. On the behavior near the origin of double sine series with monotone coefficients. Mathematica Bohemica, Tome 134 (2009) no. 3, pp. 255-273. doi: 10.21136/MB.2009.140660
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