Title: On the Growth and Approximation of Transcendental Entire Functions on Algebraic Varieties
Author(s): Devendra Kumar
Pages: 22-29
Cite as:
Devendra Kumar, On the Growth and Approximation of Transcendental Entire Functions on Algebraic Varieties, Int. J. Anal. Appl., 12 (1) (2016), 22-29.

Abstract


Let X be a complete intersection algebraic variety of codimension m > 1 in Cm+n. Inthis paper we characterized the classical growth parameters order and type for transcendental entire functions f ∈ ⊕(X), the space of holomorphic functions on the complete intersection algebraic variety X, in terms of the best polynomial approximation error in Lp-norm, 0 < p ≤ ∞, on a L − regular non-pluripolar compact subset K of Cm+n.

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References


  1. E. Bedford and B.A. Taylor, The complex equilibrium measure of a symmetric convex set in R n , Trans. Amer. Math. Soc. 294(1986), 705-717.

  2. R.P. Boas, Entire Functions, Academic Press, New York, 1954.

  3. S.M.Einstein-Matthews and H.S. Kasana, Proximate order and type of entire functions of several complex variables, Israel Journal of Mathematics 92(1995), 273-284.

  4. S.M. Einstein-Matthews and Clement H. Lutterodt, Growth of transcendental entire functions on algebraic varieties, Israel Journal of Mathematics 109(1999), 253-271.

  5. Adam Janik, On approximation of entire functions and generalized order,Univ. Iagel. Acta. Math. 24(1984), 321-326.

  6. O.P. Juneja, G.P. Kapoor and S.K. Bajpai, On the (p,q)-order and lower (p,q)-order of an entire function, J. Reine Angew. Math. 282(1976), 53-67.

  7. O.P. Juneja, G.P. Kapoor and S.K. Bajpai, On the (p,q)-type and lower (p,q)-type of an entire function, J. Reine Angew. Math. 290(1977), 180-190 .

  8. H.S. Kasana and D. Kumar, On approximation and interpolation of entire functions with index-pair (p,q), Publica- cions Matematique 38 (1994), No. 4, 681-689.

  9. D. Kumar, Generalized growth and best approximation of entire functions in L p -norm in several complex variables, Annali dell’ Universitá di Ferrara VII, 57 (2011), 353-372.

  10. P. Lelong and L. Gruman, Entire Functions of Several Complex Variables, Grundlehren der Mathematischen Wis- senschaften 282, Springer-Verlag, Berlin, 1986.

  11. B. Ja. Levin, Distributions of Zeros of Entire Functions, Translations of Mathematics Monographs 55, Amer. Math. Soc., Providence, R.I., 1994.

  12. A.R. Reddy, Approximation of an entire function, J. Approx. Theory 3(1970), 128-137.

  13. A.R. Reddy, Best polynomial approximation to certain entire functions, J. Approx. Theory 5(1972), 97-112.

  14. L.I. Ronkin, Introduction to the Theory of Entire Functions of Several Variables, Amer. Math. Soc., Providence, R.I., 1974.

  15. A. Sadullaev, Plurisubharmonic measures and capacities on complex manifolds, Russian Mathematical Surveys 36(1981), 61-119.

  16. A. Sadullaev, An estimate for polynomials on analytic sets, Mathematics of the USSR-Izvestiya 20(1980), 493-502.

  17. S.M. Shah, Polynomial approximation of an entire function and generalized orders, J. Approx. Theory 19(1977), 315-324.

  18. J. Siciak, Extremal plurisubharmonic functions and capacities in C n , Lectures in Mathematics 14, Sofia University, Tokyo, 1982.

  19. J. Siciak, On some extremal functions and their applications in the theory of analytic functions of several complex variables, Trans. Amer. Math. Soc. 105(1962), 322-357.

  20. G.S. Srivastava and Susheel Kumar, On approximation and generalized type of entire functions of several complex variables, European J. Pure Appl. Math. 2, no. 4(2009), 520-531.

  21. T.N. Winiarski, Application of approximation and interpolation methods to the examination of entire functions of n complex variables, Ann. Polon. Math. 28(1973), 97-121.

  22. T.N. Winiarski, Approximation and interpolation of entire functions, Ann. Polon. Math. 23(1970), 259-273.

  23. A. Zeriahi, Meilleure approximation polynomiale et croissante des fonctions entieres sur certaines varievalgebriques affines, Annales Inst. Fourier (Grenoble) 37(1987), 79-104.