Growth and approximation of solutions to a class of certain linear partial differential equations in ℝN

2014 ◽  
Vol 64 (1) ◽  
Author(s):  
Devendra Kumar
Keyword(s):  
Symmetric Solution ◽  
Entire Functions ◽  
Polynomial Interpolation ◽  
Growth Parameters ◽  
Harmonic Polynomial ◽  
Axially Symmetric ◽  
Harmonic Polynomials ◽  
Linear Partial Differential Equations ◽  
Order And Type ◽  
Interpolation Polynomials

AbstractIn this paper we consider the equation ∇2 φ + A(r 2)X · ∇φ + C(r 2)φ = 0 for X ∈ ℝN 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 Λn*(x, η;C n) and Λn(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 ℝN, Trans. Amer. Math. Soc. 196 (1974), 385–402] and [MARDEN, M.: Value distribution of harmonic polynomials in several real variables, Trans. Amer. math. Soc. 159 (1971), 137–154].

scholarly journals Asymptotics of Polynomial Interpolation and the Bernstein Constants

2021 ◽  
Vol 76 (2) ◽  
Author(s):  
Michael Revers
Keyword(s):  
Exponential Type ◽  
Chebyshev Polynomials ◽  
Lagrange Interpolation ◽  
Entire Functions ◽  
Polynomial Interpolation ◽  
Asymptotic Bounds ◽  
Functions Of Exponential Type ◽  
Chebyshev Interpolation ◽  
Lagrange Interpolation Polynomials ◽  
Interpolation Polynomials

AbstractIt is well known that the interpolation error for $$\left| x\right| ^{\alpha },\alpha >0$$ x α , α > 0 in $$L_{\infty }\left[ -1,1\right] $$ L ∞ - 1 , 1 by Lagrange interpolation polynomials based on the zeros of the Chebyshev polynomials of first kind can be represented in its limiting form by entire functions of exponential type. In this paper, we establish new asymptotic bounds for these quantities when $$\alpha $$ α tends to infinity. Moreover, we present some explicit constructions for near best approximation polynomials to $$\left| x\right| ^{\alpha },\alpha >0$$ x α , α > 0 in the $$L_{\infty }$$ L ∞ norm which are based on the Chebyshev interpolation process. The resulting formulas possibly indicate a general approach towards the structure of the associated Bernstein constants.

scholarly journals Best ${L^p}$-approximation of generalized biaxially symmetric potentials over Carath'eodory domains in ${C^N}$ having slow growth

2002 ◽  
Vol 33 (3) ◽  
pp. 223-232
Author(s):  
Devendra Kumar
Keyword(s):  
Function Theory ◽  
Slow Growth ◽  
Entire Functions ◽  
Growth Parameters ◽  
Approximation Error ◽  
Operator Method ◽  
Holomorphic Functions ◽  
Decomposition Theorem ◽  
Harmonic Polynomials ◽  
Classical Function

Let $ F$ be a real valued generalized biaxially symmetric potentials (GBASP) defined on the Caratheodory domain on $ C^N$. Let $ L_\mu^p(D)$ be the class of all functions $ F$ holomorphic on $ D$ such that $ \parallel F\parallel_{D,p}=[\int_D\mid F\mid^pd\mu]^{1\over p}$. Where $ \mu$ is the positive finite, Boral measure with regular asymptotic distribution on $ C^N$. For $ F\in L_{\mu}^p(D)$, set $ E_n^p(F)=\inf\{\parallel F-P\parallel_{D,p}:P\in H_n\}$, $ H_n$ consist of all real biaxisymmetric harmonic polynomials of degree at most $ 2n$. The paper deals with the growth of entire function GBASP in terms of approximation error in $ L_{\mu}^p$-norm on $ D$. The analysis utilizes the Bergman and Gilbert integral operator method to extend results from classical function theory on the best polynomial approximation of analytic functions of several complex variables. Finally we prove a generalized decomposition theorem in a new way. The paper is the generalization of the concepts of generalized growth parameters to entire functions on Caratheodory domains on $ C^N$ (instead of entire holomorphic functions on $ C$) for slow growth.

scholarly journals Best linear approximation and coefficients characterization of entire functions in doubly connected domains

Filomat ◽  
2019 ◽  
Vol 33 (15) ◽  
pp. 4827-4835
Author(s):  
Rifaqat Ali ◽  
Devendra Kumar ◽  
Mohamed Altanji
Keyword(s):  
Entire Function ◽  
Linear Approximation ◽  
Entire Functions ◽  
Growth Parameters ◽  
Series Expansions ◽  
Faber Series ◽  
Approximation Errors ◽  
Order And Type

In the present paper, we established the relations between growth parameters order and type in terms of coefficients occurring in generalized Faber series expansions of entire function and corresponding best linear approximation errors in supnorm in doubly connected domains.

Harmonic Polynomials Associated With Reflection Groups

2000 ◽  
Vol 43 (4) ◽  
pp. 496-507 ◽  
Author(s):  
Yuan Xu
Keyword(s):  
Weight Function ◽  
Homogeneous Polynomial ◽  
Harmonic Polynomial ◽  
Reflection Group ◽  
Reflection Groups ◽  
Harmonic Polynomials ◽  
Adjoint Operators

AbstractWe extend Maxwell’s representation of harmonic polynomials to h-harmonics associated to a reflection invariant weight function hk. Let 𝑫i, 1 ≤ i ≤ d, be Dunkl’s operators associated with a reflection group. For any homogeneous polynomial P of degree n,we prove the polynomial is a h-harmonic polynomial of degree n, where γ = ∑ki and 𝑫 = (𝑫1, … ,𝑫d). The construction yields a basis for h-harmonics. We also discuss self-adjoint operators acting on the space of h-harmonics.

scholarly journals Measures of Growth of Entire Solutions of Generalized Axially Symmetric Helmholtz Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-6
Author(s):  
Devendra Kumar ◽  
Rajbir Singh
Keyword(s):  
Helmholtz Equation ◽  
Lower Order ◽  
Entire Solution ◽  
Axially Symmetric ◽  
Entire Solutions ◽  
Lower Type ◽  
Order And Type

For an entire solution of the generalized axially symmetric Helmholtz equation , measures of growth such as lower order and lower type are obtained in terms of the Bessel-Gegenbauer coefficients. Alternative characterizations for order and type are also obtained in terms of the ratios of these successive coefficients.

scholarly journals An axially symmetric solution of metric-affine gravity

1996 ◽  
Vol 13 (12) ◽  
pp. 3253-3259 ◽  
Author(s):  
E J Vlachynsky ◽  
R Tresguerres ◽  
Yu N Obukhov ◽  
F W Hehl
Keyword(s):  
Symmetric Solution ◽  
Axially Symmetric ◽  
Affine Gravity
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