On simple exponential sets of polynomials. (English) Zbl 1290.32002

Kishka, Z. G.; Saleem, M. A.; Abul-Dahab, M. A.

Mediterr. J. Math. 11, No. 2, 337-347 (2014).

Summary: We show that certain classes of regular functions of several complex variables can be represented by exponential sets of polynomials in hyperelliptic regions. Moreover, an upper bound for the order of the exponential set is given.

MSC:

32A05	Power series, series of functions of several complex variables
32A17	Special families of functions of several complex variables
32A99	Holomorphic functions of several complex variables

Keywords:

basic sets of polynomials; exponential sets; hyperelliptic regions

References:

[1]	Abul-Ez M.: Inverse sets of polynomials in Clifford analysis. Arch. der Math. 58, 561–567 (1992) · Zbl 0739.30037
[2]	Abul-Ez M., Constales D.: Similar functions and similar bases of polynomials in Clifford setting. Complex Variables 48, 1055–1070 (2003) · Zbl 1052.30048
[3]	Abul-Ez, M., Saleem, M., Zayed, M.: On the representation near a point of Clifford valued functions by infinite series of polynomials. In: 9th International Conferenceon Clifford Algebras, Weimar, Germany, 15–20 July (2011)
[4]	Abul-Ez M., Morais J., Zayed M.: Generalized derivative and primitive of Cliffordian bases of polynomials constructed through Appell monomials. Comput. Meth. Func. Theo. 4, 501–515 (2012) · Zbl 1257.30058
[5]	Abul-Ez M., Saleem M., Abd-Elmageed H., Morais J.: On polynomial series expansions of Cliffordian functions in open balls and at the origin. Cliff. Anal. Cliff. Alge. Appl. 2, 291–307 (2012)
[6]	Abul-Ez M., Constales D.: On the order of basic series representing Clifford valued functions. Appl. Math. Comput. 142, 575–584 (2003) · Zbl 1058.30040
[7]	Cannon B.: On the convergence of series of polynomials. Proc. Lond. Math. Soc. 43, 348–364 (1937) · Zbl 0017.25301
[8]	Cannon B.: On the representation of integral functions by general basic series. Math. Zeit. 45, 185–208 (1939) · Zbl 0021.03801
[9]	El-Sayed A.: On derived and integrated sets of basic sets of polynomials of several complex variables. Acta Math. Acad. Paed. Nyiregyhazi 19, 195–204 (2003) · Zbl 1048.32001
[10]	El-Sayed, A., Kishka, Z.M.G.: On the effectiveness of basic sets of polynomials of several complex variables in elliptical regions. In: Proceedings of the 3rd International ISAAC Congress, pp. 265-278. Freie Universitaet Berlin, Germany. Kluwer, Dordrecht (2003) · Zbl 1152.32300
[11]	Hassan G.F.: Ruscheweyh differential operator sets of basic sets of polynomials of several complex variables in hyperelliptical regions. Acta Math. Acad. Paed. Nyiregyhazi 22, 247–264 (2006) · Zbl 1120.32001
[12]	Kishka Z.M.G.: On the convergence of properties of basic sets of polynomials of several complex variables i. Sohg Pure, Appl. Sci. Bull. Fac. Sci. Assiut Univ. 7, 71–119 (1991)
[13]	Kishka Z.M.G.: Power set of simple set of polynomials of two complex variables. Bull. Soc. R. Sci. Liege 62, 265–278 (1993) · Zbl 0804.32002
[14]	Kishka Z.M.G, El-Sayed A.: On the order and type of basic and composite sets of polynomials in complete Reinhardt domains. Period. Math. Hung. 46, 67–79 (2003) · Zbl 1027.32002
[15]	Mursi, M., Maker, B.H.: Basic sets of polynomials of several complex variables i. In: The Second Arab Sci. Congress, pp. 51–60. Cairo (1955)
[16]	Mursi, M., Maker, B.H.: Basic sets of polynomials of several complex variables ii. In: The Second Arab Sci. Congress, pp. 61–68, Cairo (1955)
[17]	Nassif M.: Composite sets of polynomials of several complex variables. Publ. Math. Debrecen 18, 43–52 (1971) · Zbl 0247.32004
[18]	Sayyed K.A.M., Metwally M.S., Hassan G.F.: Effectiveness of equivalent sets of polynomials of two complex variables in polycylinders and Faber regions. Int. J. Math. Math. Sci. 2, 529–539 (2000) · Zbl 1171.32301
[19]	Whittaker J.M.: On series of polynomials. Quart. J. Math. Oxford 5, 224–239 (1934) · Zbl 0009.34205
[20]	Whittaker, J.M.: Sur les series de base de polynomes quelconques. Gauthier-Villars, Paris (1949) · Zbl 0038.22804