# On a Difference Equation Generated by Two “Close” Squares

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## Abstract

We consider a set of two squares constructed for the primitive periods 1 and *i* and having four vertices on one straight line. In a neighborhood of this set, we study a four-element difference equation with constant coefficients. The linear shifts of this equation are the generating transformations of the corresponding doubly periodic group and their inverse transformations. The solution is sought in the class of functions that are analytic outside this set and vanish at infinity. We give some applications to the moment problem for entire functions of exponential type.

## Keywords and phrases

difference equations method of regularization moment problem## Preview

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## References

- 1.V. V. Napalkov,
*Convolution Equations in Many–Dimensional Spaces*(Nauka, Moscow, 1988) [in Russian].Google Scholar - 2.F. N. Garif’yanov, “Difference equations for functions analytic beyond several squares,” Sib. Mat. Zh.
**44**, 435–442 (2003).CrossRefzbMATHGoogle Scholar - 3.F. N. Garif’yanov and S. A. Modina, “On the four–element equation for the functions analytic beyond a trapezoid and its applications,” Sib. Math. J.
**52**, 191–196 (2011). doi 10.1134/S0037446611020017MathSciNetCrossRefzbMATHGoogle Scholar - 4.F. N. Garif’yanov, “Inversion problem for singular integral and difference equations for functions analytical outside a square,” Russ. Math.
**37**(7), 5–14 (1993).MathSciNetzbMATHGoogle Scholar - 5.F. N. Garif’yanov, “Regularization of one class of difference equations,” Sib. Math. J.
**42**, 846–850 (2001).CrossRefzbMATHGoogle Scholar - 6.F. N. Garif’yanov and E. V. Nasyrova, “Regularization of linear difference equations with analytic coefficients and their applications,” Russ. Math. (Iz. VUZ)
**55**(11), 66–70 (2011).MathSciNetCrossRefzbMATHGoogle Scholar - 7.F. N. Garif’yanov, “Stieltjes moments of entire functions of exponential type,” Math. Notes
**67**, 572–576 (2000). doi 10. 1007/BF02676327MathSciNetCrossRefzbMATHGoogle Scholar - 8.F. N. Garif’yanov, “On a certain difference equation and its application to the moment problem,” Math. Notes
**73**, 777–782 (2003).MathSciNetCrossRefzbMATHGoogle Scholar - 9.F. N. Garif’yanov and E. V. Strezhneva, “A difference equation for functions analytic outside two squares,” Russ. Math. (Iz. VUZ)
**62**(6), 3–8 (2018).zbMATHGoogle Scholar - 10.L. I. Chibrikova, “Boundary–value problems for a rectangle,” Uch. Zap. Kazansk. Univ.
**123**(9), 15–39 (1963).MathSciNetGoogle Scholar - 11.E. P. Aksent’eva and F. N. Garif’yanov, “On the investigation of an integral equation with a Carleman kernel,” Sov. Math. (Iz. VUZ)
**27**(4), 53–63 (1983).MathSciNetzbMATHGoogle Scholar - 12.L. Bieberbach,
*Analytische Fortsetzung*(Springer, Berlin, 1955).CrossRefzbMATHGoogle Scholar

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