On the behavior near the origin of double sine series with monotone coefficients

Xhevat Z. Krasniqi

Mathematica Bohemica (2009)

  • Volume: 134, Issue: 3, page 255-273
  • ISSN: 0862-7959

Abstract

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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 ) ∼ m n a m , n + m n ∑ l = 1 n - 1 l a m , l + n m ∑ k = 1 m - 1 k a k , n + 1 m n ∑ l = 1 n - 1 ∑ k = 1 m - 1 k l a k , l , where x ∈ ( π m + 1 , π m ] , y ∈ ( π n + 1 , π n ] , m , n = 1 , 2 , ⋯ .

How to cite

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Krasniqi, Xhevat Z.. "On the behavior near the origin of double sine series with monotone coefficients." Mathematica Bohemica 134.3 (2009): 255-273. <http://eudml.org/doc/38091>.

@article{Krasniqi2009,
abstract = {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 \[ g(x,y)\sim mna\_\{m,n\}+\frac\{m\}\{n\}\sum \_\{l=1\}^\{n-1\}la\_\{m,l\}+\frac\{n\}\{m\}\sum \_\{k=1\}^\{m-1\}ka\_\{k,n\}+\frac\{1\}\{mn\}\sum \_\{l=1\}^\{n-1\}\sum \_\{k=1\}^\{m-1\}kla\_\{k,l\}, \] where $x\in (\frac\{\pi \}\{m+1\}, \frac\{\pi \}\{m\}]$, $ y\in (\frac\{\pi \}\{n+1\}, \frac\{\pi \}\{n\}]$, $ m, n=1,2,\dots $.},
author = {Krasniqi, Xhevat Z.},
journal = {Mathematica Bohemica},
keywords = {double sine series; sum of a double sine series with monotone coefficients; double sine series; sum of a double sine series with monotone coefficients},
language = {eng},
number = {3},
pages = {255-273},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the behavior near the origin of double sine series with monotone coefficients},
url = {http://eudml.org/doc/38091},
volume = {134},
year = {2009},
}

TY - JOUR
AU - Krasniqi, Xhevat Z.
TI - On the behavior near the origin of double sine series with monotone coefficients
JO - Mathematica Bohemica
PY - 2009
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 134
IS - 3
SP - 255
EP - 273
AB - 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 \[ g(x,y)\sim mna_{m,n}+\frac{m}{n}\sum _{l=1}^{n-1}la_{m,l}+\frac{n}{m}\sum _{k=1}^{m-1}ka_{k,n}+\frac{1}{mn}\sum _{l=1}^{n-1}\sum _{k=1}^{m-1}kla_{k,l}, \] where $x\in (\frac{\pi }{m+1}, \frac{\pi }{m}]$, $ y\in (\frac{\pi }{n+1}, \frac{\pi }{n}]$, $ m, n=1,2,\dots $.
LA - eng
KW - double sine series; sum of a double sine series with monotone coefficients; double sine series; sum of a double sine series with monotone coefficients
UR - http://eudml.org/doc/38091
ER -

References

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  1. Young, W. H., On the mode of oscillation of a Fourier series and of its allied series, Proc. London Math. Soc. 12 (1913), 433-452 4.0301.02. (1913) 
  2. Salem, R., Détermination de l'ordre de grandeur à l'origine de certains séries trigonométriques, C. R. Acad. Sci. Paris 186 (1928), 1804-1806 4.0313.01. (1928) 
  3. Salem, R., Essais sur les séries trigonométriques, Paris (1940) 6.1235.02. Zbl0027.20901MR0002656
  4. Hartman, Ph., Wintner, A., 10.1112/jlms/s1-28.1.102, J. London Math. Soc. 28 (1953), 102-104. (1953) Zbl0050.07206MR0051959DOI10.1112/jlms/s1-28.1.102
  5. Shogunbekov, Sh. Sh., Certain estimates for sine series with convex coefficients, Russian Primenenie Funkcional'nogo analiza v teorii priblizhenii, Tver' (1993), 67-72. (1993) 
  6. Aljančić, S., Bojanić, R., Tomić, M., Sur le comportement asymptotique au voisinage de zéro des séries trigonométriques de sinus à coefficients monotones, Publ. Inst. Math. Acad. Serie Sci. 10 (1956), 101-120. (1956) MR0082579
  7. Krasniqi, Xh. Z., Braha, N. L., On the behavior of r -th derivative near the origin of sine series with convex coefficients, J. Inequal. Pure Appl. Math. 8 (2007), 1, Paper No. 22 (electronic only), 6 pp., http://jipam.vu.edu.au. (2007) MR2295716
  8. Telyakovskiǐ, S. A., On the behavior near the origin of the sine series with convex coefficients, Publ. Inst. Math. Nouvelle Sér. 58 (1995), 43-50. (1995) MR1396113
  9. Popov, A. Yu., 10.1023/B:MATN.0000009019.66625.fb, Mathematical Notes 74 (2003), 829-840. (2003) Zbl1156.42303MR2054006DOI10.1023/B:MATN.0000009019.66625.fb
  10. Hardy, H. G., On double Fourier series, and especially those which represent the double zeta-function with real and incommensurable parameters, Quarterly J. Math. 37 (1906), 53-79 (Collected Papers: Vol. IV, pp. 433-459). (1906) 
  11. Wittaker, E. T., Watson, G. N., A course of modern analysis I, Nauka Moskva (1963), Russian. (1963) 
  12. Vukolova, T. M., Dyachenko, M. I., Bounds for norms of sums of double trigonometric series with multiply monotone coefficients, Russ. Math. (1994), 38 18-26. (1994) MR1317218
  13. Vukolova, T. M., Dyachenko, M. I., On the properties of sums of trigonometric series with monotone coefficients, Mosc. Univ. Math. Bull. (1995), 50 19-27. (1995) Zbl0881.42004MR1376350

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