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 On Bleimann-Butzer-Hahn operators for exponential functions

2007 ◽  
Vol 75 (3) ◽  
pp. 409-415 ◽  
Author(s):  
Ulrich Abel ◽  
Mircea Ivan
Keyword(s):  
Exponential Type ◽  
Binomial Coefficients ◽  
Monotone Functions ◽  
Exponential Functions ◽  
Approximation Process

Some inequalities involving the binomial coefficients are obtained. They are used to determine the domain of convergence of the Bleimann, Butzer and Hahn approximation process for exponential type functions. An answer to Hermann's conjecture related to the Bleimann, Butzer and Hahn operators for monotone functions is 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 On Berman's phenomenon in interpolation theory

1975 ◽  
Vol 12 (3) ◽  
pp. 457-465 ◽  
Author(s):  
W. Lyle Cook ◽  
T.M. Mills
Keyword(s):  
Higher Order ◽  
Interpolation Process ◽  
Divergence Theorem ◽  
Interpolation Theory ◽  
Open Problems ◽  
Chebyshev Nodes

In 1965, D.L. Berman established an interesting divergence theorem concerning Hermite-Fejér interpolation on the extended Chebyshev nodes. In this paper it is shown that this phenomenon is not an isolated incident. A similar divergence theorem is proved for a higher order interpolation process. The paper closes with a list of several related open problems.

scholarly journals On convergence and divergence of Fourier expansions associated to Jacobi measure with mass points

Filomat ◽  
2009 ◽  
Vol 23 (1) ◽  
pp. 61-68
Author(s):  
Bujar Fejzullahu
Keyword(s):  
Fourier Expansion ◽  
Necessary Conditions ◽  
Delta Function ◽  
Orthonormal Polynomials ◽  
Mean Convergence ◽  
Fourier Expansions ◽  
Convergence And Divergence

We prove the failure of a.e. convergence of the Fourier expansion in terms of the orthonormal polynomials with respect to the measure (1 - x)?(1 + x)?dx + M?1 + N?1, where ?t is the delta function at a point t and M > 0; N > 0: Lebesgue norms of Koornwinder's Jacobi-type polynomials are applied to obtain a new proof of necessary conditions for mean convergence. .

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