Analysis of the Bernstein basis functions: an approach to combinatorial sums involving binomial coefficients and Catalan numbers

2014 ◽  
Vol 38 (14) ◽  
pp. 3007-3021 ◽  
Author(s):  
Yilmaz Simsek
Keyword(s):  
Catalan Numbers ◽  
Basis Functions ◽  
Binomial Coefficients ◽  
Bernstein Basis ◽  
Combinatorial Sums ◽  
Bernstein Basis Functions

scholarly journals Combinatorial identities associated with Bernstein type basis functions

Filomat ◽  
2016 ◽  
Vol 30 (7) ◽  
pp. 1683-1689 ◽  
Author(s):  
Yilmaz Simsek
Keyword(s):  
Catalan Numbers ◽  
Combinatorial Identities ◽  
Basis Functions ◽  
Binomial Coefficients ◽  
Bernstein Basis ◽  
Bernstein Type ◽  
Combinatorial Sums ◽  
Bernstein Basis Functions

In this paper, we give some identities and relations for the Bernstein basis functions and the beta type polynomials. Integrating these identities, we derive many identities and formulas, some old and some new, for combinatorial sums involving binomial coefficients and the Catalan numbers. We also give remarks and comments on these identities.

scholarly journals Using Bezier curves in medical applications

Filomat ◽  
2016 ◽  
Vol 30 (4) ◽  
pp. 937-943 ◽  
Author(s):  
Buket Simsek ◽  
Ahmet Yardimci
Keyword(s):  
Cross Sections ◽  
Biomedical Applications ◽  
Medical Physics ◽  
Binomial Coefficients ◽  
Medical Applications ◽  
Bernstein Basis ◽  
Bezier Curves ◽  
Bézier Curves ◽  
Combinatorial Sums ◽  
Bernstein Basis Functions

In this paper we survey the 3D reconstruction of an object from its 2D cross-sections has many applications in different fields of sciences such as medical physics and biomedical applications. The aim of this paper is to give not only the Bezier curves in medical applications, but also by using generating functions for the Bernstein basis functions and their identities, some combinatorial sums involving binomial coefficients are deriven. Finally, we give some comments related to the above areas.

8-node quasi-conforming plane element by using Bernstein basis functions

2018 ◽  
Vol 70 ◽  
pp. 127-140 ◽  
Author(s):  
Changsheng Wang ◽  
Xingtong Lu ◽  
Xiangkui Zhang ◽  
Ping Hu
Keyword(s):  
Basis Functions ◽  
Bernstein Basis ◽  
Plane Element ◽  
Bernstein Basis Functions

scholarly journals Decomposition of mixed pixels in MODIS data using Bernstein basis functions

2019 ◽  
Vol 13 (04) ◽  
pp. 1
Author(s):  
Yi Qin ◽  
Feng Guo ◽  
Yupeng Ren ◽  
Xin Wang ◽  
Juan Gu ◽  
...  
Keyword(s):  
Basis Functions ◽  
Bernstein Basis ◽  
Modis Data ◽  
Mixed Pixels ◽  
Bernstein Basis Functions

scholarly journals On a New Construction of Generalized q-Bernstein Polynomials Based on Shape Parameter λ

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1919
Author(s):  
Qing-Bo Cai ◽  
Reşat Aslan
Keyword(s):  
Error Estimation ◽  
Shape Parameter ◽  
Bernstein Polynomials ◽  
Modulus Of Smoothness ◽  
Basis Functions ◽  
Approximation Properties ◽  
Bernstein Basis ◽  
New Class ◽  
New Construction ◽  
Bernstein Basis Functions

This paper deals with several approximation properties for a new class of q-Bernstein polynomials based on new Bernstein basis functions with shape parameter λ on the symmetric interval [−1,1]. Firstly, we computed some moments and central moments. Then, we constructed a Korovkin-type convergence theorem, bounding the error in terms of the ordinary modulus of smoothness, providing estimates for Lipschitz-type functions. Finally, with the aid of Maple software, we present the comparison of the convergence of these newly constructed polynomials to the certain functions with some graphical illustrations and error estimation tables.

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