scholarly journals Derivatives of Orthonormal Polynomials and Coefficients of Hermite-Fejér Interpolation Polynomials with Exponential-Type Weights

2010 ◽  
Vol 2010 ◽  
pp. 1-29 ◽  
Author(s):  
H. S. Jung ◽  
R. Sakai
Keyword(s):  
Exponential Type ◽  
Orthonormal Polynomials ◽  
Interpolation Polynomials ◽  
Derivatives Of

scholarly journals Convergence and Divergence of Higher-Order Hermite or Hermite-Fejér Interpolation Polynomials with Exponential-Type Weights

2012 ◽  
Vol 2012 ◽  
pp. 1-31
Author(s):  
Hee Sun Jung ◽  
Gou Nakamura ◽  
Ryozi Sakai ◽  
Noriaki Suzuki
Keyword(s):  
Exponential Type ◽  
Higher Order ◽  
Divergence Theorem ◽  
Convergence Theorems ◽  
Approximation Process ◽  
Orthonormal Polynomials ◽  
Convergence And Divergence ◽  
Interpolation Polynomials

Let and let , where and is an even function. Then we can construct the orthonormal polynomials of degree for . In this paper for an even integer we investigate the convergence theorems with respect to the higher-order Hermite and Hermite-Fejér interpolation polynomials and related approximation process based at the zeros of . Moreover, for an odd integer , we give a certain divergence theorem with respect to the higher-order Hermite-Fejér interpolation polynomials based at the zeros of .

The Sector Problem

1957 ◽  
Vol 24 (4) ◽  
pp. 574-581
Author(s):  
G. Horvay ◽  
K. L. Hanson
Keyword(s):  
Variational Method ◽  
Orthogonal Polynomials ◽  
Biharmonic Equation ◽  
Approximate Solutions ◽  
Circular Sector ◽  
Orthonormal Polynomials ◽  
Circular Boundary ◽  
Stress Functions ◽  
Complete Set ◽  
Derivatives Of

Abstract On the basis of the variational method, approximate solutions f k ( r ) h k ( θ ) , f k ( r ) g k ( θ ) , F k ( θ ) H k ( r ) , F k ( θ ) G k ( r ) of the biharmonic equation are established for the circular sector with the following properties: The stress functions fkhk create shear tractions on the radial boundaries; the stress functions fkgk create normal tractions on the radial boundaries; the stress functions FkHk create both shear and normal tractions on the circular boundary, and the stress functions FkGk create normal tractions on the circular boundary. The enumerated tractions are the only tractions which these function sets create on the various boundaries of the sector. The factors fk(r) constitute a complete set of orthonormal polynomials in r into which (more exactly, into the derivatives of which) self-equilibrating normal or shear tractions applied to the radial boundaries of the sector may be expanded; the factors Fk(θ) constitute a complete set of orthonormal polynomials in θ into which shear tractions applied to the circular boundary of the sector may be expanded; and the functions Fk″ + Fk constitute a complete set of non-orthogonal polynomials into which normal tractions applied to the circular boundary of the sector may be expanded. Function tables, to facilitate the use of the stress functions, are also presented.

scholarly journals Eksponencijalne atomske bazne funkcije : razvoj i primjena

2016 ◽  
Author(s):  
◽  
Nives Brajčić Kurbaša
Keyword(s):  
Exponential Type ◽  
Function Approximation ◽  
Numerical Solutions ◽  
Basis Functions ◽  
Practical Application ◽  
Software Module ◽  
Practical Advantage ◽  
First Time ◽  
Derivatives Of ◽  
Basic Level

In this work basic properties of algebraic Atomic Basis Functions (ABF) are systematized and, using analogous approach, ABF of exponential type, so far known only at the basic level, are developed. For the first time the properties of exponential ABFs are thoroughly investigated and expressions for calculating the values and all the necessary derivatives of the functions in an arbitrary points of the domain are developed as well as some special features required for their practical application in a form suitable for numerical analysis. A software module for calculating all necessary values of the exponential ABFs, including its own graphics support, is created within this work. Thus, the exponential ABF is prepared to use as users or compiler function. The presented 1D verification examples of the function approximation and the examples of solving differential equations illustrate and confirm the practical advantage of the ABFs of the exponential type in relation to the, so far mostly used, algebraic functions, especially for describing expressed fronts and/or waves contained in the numerical solutions of various technical tasks.

scholarly journals Exponential moments and simultaneous approximation properties for Durrmeyer type operators with weights of Szasz basis functions

Filomat ◽  
2021 ◽  
Vol 35 (5) ◽  
pp. 1465-1475
Author(s):  
Antonio-Jesús López-Moreno ◽  
Vijay Gupta
Keyword(s):  
Exponential Type ◽  
Exponential Growth ◽  
Simultaneous Approximation ◽  
Basis Functions ◽  
Approximation Properties ◽  
Exponential Functions ◽  
Quantitative Estimates ◽  
Exponential Moments ◽  
Durrmeyer Type Operators ◽  
Derivatives Of

The present paper deals with the approximation properties for exponential functions of general Durrmeyer type operators having the weights of Sz?sz basis functions. Here we give explicit expressions for exponential type moments by means of which we establish, for the derivatives of the operators, the Voronovskaja formulas for functions of exponential growth and the corresponding weighted quantitative estimates for the remainder in simultaneous approximation.

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