scholarly journals Lebesgue Function for Higher Order Hermite-Fej´er Interpolation Polynomials with Exponential-Type Weights

2020 ◽  
Vol 5 (4) ◽  
Author(s):  
Ryozi SAKAI ◽  
Keyword(s):  
Exponential Type ◽  
Higher Order ◽  
Lebesgue Function ◽  
Interpolation Polynomials

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 .

Weighted higher order exponential type inequalities in metric spaces and applications

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Huiju Wang ◽  
Pengcheng Niu
Keyword(s):  
Homogeneous Space ◽  
Lie Groups ◽  
Sobolev Spaces ◽  
Exponential Type ◽  
Metric Spaces ◽  
Type Inequality ◽  
Higher Order ◽  
Geodesic Space ◽  
Stratified Lie Groups ◽  
Weighted Case

AbstractIn this paper, we establish weighted higher order exponential type inequalities in the geodesic space {({X,d,\mu})} by proposing an abstract higher order Poincaré inequality. These are also new in the non-weighted case. As applications, we obtain a weighted Trudinger’s theorem in the geodesic setting and weighted higher order exponential type estimates for functions in Folland–Stein type Sobolev spaces defined on stratified Lie groups. A higher order exponential type inequality in a connected homogeneous space is also given.

scholarly journals Impact on Stability by the Use of Memory in Traub-Type Schemes

Mathematics ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 274
Author(s):  
Francisco I. Chicharro ◽  
Alicia Cordero ◽  
Neus Garrido ◽  
Juan R. Torregrosa
Keyword(s):  
Order Of Convergence ◽  
Nonlinear Problems ◽  
Higher Order ◽  
Lower Number ◽  
Dynamical Analysis ◽  
The Real ◽  
Original Scheme ◽  
Interpolation Polynomials ◽  
With Memory ◽  
Number Of Iterations

In this work, two Traub-type methods with memory are introduced using accelerating parameters. To obtain schemes with memory, after the inclusion of these parameters in Traub’s method, they have been designed using linear approximations or the Newton’s interpolation polynomials. In both cases, the parameters use information from the current and the previous iterations, so they define a method with memory. Moreover, they achieve higher order of convergence than Traub’s scheme without any additional functional evaluations. The real dynamical analysis verifies that the proposed methods with memory not only converge faster, but they are also more stable than the original scheme. The methods selected by means of this analysis can be applied for solving nonlinear problems with a wider set of initial estimations than their original partners. This fact also involves a lower number of iterations in the process.

scholarly journals The Lebesgue function for Hermite-Fejér interpolation on the extended Chebyshev nodes

2002 ◽  
Vol 66 (1) ◽  
pp. 151-162
Author(s):  
Simon J. Smith
Keyword(s):  
Chebyshev Polynomial ◽  
Minimum Degree ◽  
Convergence Property ◽  
Lebesgue Constant ◽  
Interpolation Polynomial ◽  
Lebesgue Function ◽  
Surprising Result ◽  
End Points ◽  
Chebyshev Nodes ◽  
Interpolation Polynomials

Given f ∈ C[−1, 1] and n point (nodes) in [−1, 1], the Hermite-Fejér interpolation polynomial is the polynomial of minimum degree which agrees with f and has zero derivative at each of the nodes. In 1916, L. Fejér showed that if the nodes are chosen to be zeros of Tn (x), the nth Chebyshev polynomial of the first kind, then the interpolation polynomials converge to f uniformly as n → ∞. Later, D.L. Berman demonstrated the rather surprising result that this convergence property no longer holds true if the Chebyshev nodes are extended by the inclusion of the end points −1 and 1 in the interpolation process. The aim of this paper is to discuss the Lebesgue function and Lebesgue constant for Hermite-Fejér interpolation on the extended Chebyshev nodes. In particular, it is shown that the inclusion of the two endpoints causes the Lebesgue function to change markedly, from being identically equal to 1 for the Chebyshev nodes, to having the form 2n2(1 − x2)(Tn (x))2 + O (1) for the extended Chebyshev nodes.

scholarly journals Infinite Elements and Their Influence on Normal and Radiation Modes in Exterior Acoustics

2017 ◽  
Vol 25 (04) ◽  
pp. 1650020 ◽  
Author(s):  
Lennart Moheit ◽  
Steffen Marburg
Keyword(s):  
Degrees Of Freedom ◽  
Acoustic Radiation ◽  
Computational Cost ◽  
Normal Modes ◽  
Higher Order ◽  
Infinite Elements ◽  
Odd Order ◽  
Infinite Element Method ◽  
Interpolation Polynomials ◽  
Element Method

Acoustic radiation modes (ARMs) and normal modes (NMs) are calculated at the surface of a fluid-filled domain around a solid structure and inside the domain, respectively. In order to compute the exterior acoustic problem and modes, both the finite element method (FEM) and the infinite element method (IFEM) are applied. More accurate results can be obtained by using finer meshes in the FEM or higher-order radial interpolation polynomials in the IFEM, which causes additional degrees of freedom (DOF). As such, more computational cost is required. For this reason, knowledge about convergence behavior of the modes for different mesh cases is desirable, and is the aim of this paper. It is shown that the acoustic impedance matrix for the calculation of the radiation modes can be also constructed from the system matrices of finite and infinite elements instead of boundary element matrices, as is usually done. Grouping behavior of the eigenvalues of the radiation modes can be observed. Finally, both kinds of modes in exterior acoustics are compared in the example of the cross-section of a recorder in air. When the number of DOF is increased by using higher-order radial interpolation polynomials, different eigenvalue convergences can be observed for interpolation polynomials of even and odd order.

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.

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