2 φ + A(r 2)X · ??+ C(r 2)φ = 0 for X ??sup class="a-plus-plus"> N whose coefficients are entire functions of the variable r = |X|. Corresponding to a specified axially symmetric solution φ and set C n of (n + 1) circles, an axially symmetric solution Λ n * (x, η;C n ) and Λ n (x, η;C n ) are found that interpolates to φ(x, η) on the C n N , Trans. Amer. Math. Soc. 196 (1974), 385-02] and [MARDEN, M.: Value distribution of harmonic polynomials in several real variables, Trans. Amer. math. Soc. 159 (1971), 137-54]." />
Growth and approximation of solutions to a class of certain linear partial differential equations in ?sup class="a-plus-plus"> N
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  • 作者：Devendra Kumar (1)
  • 关键词：Primary 41A05 ; Secondary 31B05 ; 31B10 ; axisymmetric harmonic polynomials ; axi ; convex region ; order and type ; Lagrange polynomial ; approximation and interpolation errors ; Bergman operator and hypersphere
  • 刊名：Mathematica Slovaca
  • 出版年：2014
  • 出版时间：February 2014
  • 年：2014
  • 卷：64
  • 期：1
  • 页码：139-154
  • 全文大小：250 KB
  • 参考文献：1. BERGMAN, S.: / Integral Operators in the Theory of Linear Partial Differential Equations. Ergeb. Math. Grenzgeb. 23, Springer-Verlag, Inc., New York, 1969. CrossRef
    2. BERGMAN, S.: / Classes of solutions of linear partial differential equations in three variables, Duke J. Math. 13 (1946), 419-58. CrossRef
    3. GILBERT, R. P.: / Function Theoretic Methods in Partial Differential Equations. Math. in Science and Engineering, Vol. 54, Academic Press, New York, 1969. CrossRef
    4. GILBERT, R. P.-HOWARD, H. C.: / On a class of elliptic partial differential equations, Port. Math. 26 (1967), 353-73.
    5. GILBERT, R. P.-HOWARD, H. C.: / The scope of the function theoretic approach for equations permitting a separation of variables, J. Math. Anal. Appl. 34 (1971), 671-84. CrossRef
    6. GILBERT, R. P.: / Constructive Methods for Elliptic Equations. Lecture Notes in Math. 365, Springer Verlag, Inc., New York, 1974.
    7. HILLE, E.: / Analytic Function Theory, Vol. 2, Blaisdell, Waltham, MASS. 1962.
    8. ISAMUKHAMEDOV, S. S.-ORAMOV, ZH.: / Boundary value problems for an equation of mixed type of the second kind with non-smooth degeneration lines, Differ. Uravn. 18 (1982), 324-34 [Differ. Equ. 18 (1982), 263-71].
    9. KAPOOR, G. P.-NAUTIYAL, A.: / On the growth of generalized axisymmetric potentials, Indian J. Pure Appl. Math. 13 (1982), 1240-245.
    10. KHE KEN CHER: / On the uniqueness of solution of the Tricomi problem for equations with two lines of degenracy, Partial Differential Euqaitons, Inst. Mat. Sibirsk Otdel, Akad. Nauk. SSSR, Novosibissk, 1980, 64-7 (Russian).
    11. KUMAR, D.: / Approximation of generalized axisymmetric potentials having fast growth, Indian J. Math. 45 (2003), 265-78.
    12. KUMAR, D.-ARORA, K. N.: / Growth and approximation properties of generalized axysymmetric potentials, Demonstratio Math. XLIII (2010), 107-16.
    13. MARDEN, M.: / Axisymmetric harmonic interpolation polynomials in RN, Trans. Amer. Math. Soc. 196 (1974), 385-02.
    14. MARDEN, M.: / Value distribution of harmonic polynomials in several real variables, Trans. Amer. math. Soc. 159 (1971), 137-54.
    15. MARICHEV, O. I.: / Boundary value problems for equations of mixed type with two lines of degenracy, Izv. Akad. Nauk Beloruss. SSR Ser. Fiz-Mat. Nauk 5 (1970), 21-9 (Russian).
    16. MCCOY, P. A.: / On the zeros of generalized axisymmetric potentials, Proc. Amer. Math. Soc. 61 (1976), 54-8. CrossRef
    17. MCCOY, P. A.: / Polynomial approximation and growth of generalized axisymmetric potentials, Cand. J. Math. 31 (1979), 40-9.
    18. MCCOY, P. A.: / Interpolation and approximation of solutions to a class of linear partial differential equations in several real variables, Complex Var. Elliptic Equations 26 (1994), 213-23. CrossRef
    19. SRIVASTAVA, G. S.: / Approximation and growth of generalized axisymmetric potentials, Approx. Theory Appl. (N.S.) 12 (1996), 96-04.
    20. SZEGO, G.: / Orthogonal Polynomials (2nd ed.). Amer. Math. Soc. Colloq. Publ. 23, Amer. Math. Soc., Providence, RI, 1959.
    21. TERSENOV, S. A.: / Basic boundary value problems for one ultra parabolic equation, Sibirsk. Mat. Zh. 42 (2001), 1413-430.
    22. TERSENOV, S. A.: / Cauchy problem for a system of equations of ultraparabolic type, Mat. Zametki 77 (2005), 768-74. CrossRef
    23. WINIARSKI, T. N.: / Application of approximation and interpolation methods to the examination of entire functions of n complex variables, Ann. Polon. Math. 28 (1973), 97-21.
  • 作者单位：Devendra Kumar (1)

    1. Department of Mathematics, M.M.H. College, Model Town Ghaziabad, 201 001, U.P., India
  • ISSN：1337-2211
文摘
In this paper we consider the equation ?sup class="a-plus-plus">2 φ + A(r 2)X · ??+ C(r 2)φ = 0 for X ??sup class="a-plus-plus"> N whose coefficients are entire functions of the variable r = |X|. Corresponding to a specified axially symmetric solution φ and set C n of (n + 1) circles, an axially symmetric solution Λ n * (x, η;C n ) and Λ n (x, η;C n ) are found that interpolates to φ(x, η) on the C n and converges uniformly to φ(x, η) on certain axially symmetric domains. The main results are the characterization of growth parameters order and type in terms of axially symmetric harmonic polynomial approximation errors and Lagrange polynomial interpolation errors using the method developed in [MARDEN, M.: Axisymmetric harmonic interpolation polynomials in ?sup class="a-plus-plus"> N , Trans. Amer. Math. Soc. 196 (1974), 385-02] and [MARDEN, M.: Value distribution of harmonic polynomials in several real variables, Trans. Amer. math. Soc. 159 (1971), 137-54].

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