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On the behavior near the origin of double sine series with monotone coefficients. (English) Zbl 1212.42006

Summary: 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 \[ g(x,y)\sim mna_{m,n}+\frac {m}{n}\sum _{l=1}^{n-1}la_{m,l}+\frac {n}{m}\sum _{k=1}^{m-1}ka_{k,n}+\frac {1}{mn}\sum _{l=1}^{n-1}\sum _{k=1}^{m-1}kla_{k,l}, \] where \(x\in (\frac {\pi }{m+1}, \frac {\pi }{m}]\), \( y\in (\frac {\pi }{n+1}, \frac {\pi }{n}]\), \( m, n=1,2,\dots \).

MSC:

42A20 Convergence and absolute convergence of Fourier and trigonometric series
42A16 Fourier coefficients, Fourier series of functions with special properties, special Fourier series
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