scholarly journals Analysis and construction of a family of refinable functions based on generalized Bernstein polynomials

2016 ◽  
Vol 2016 (1) ◽  
Author(s):  
Ting Cheng ◽  
Xiaoyuan Yang
Keyword(s):  
Bernstein Polynomials ◽  
Refinable Functions ◽  
Generalized Bernstein Polynomials

scholarly journals Compactly Supported Tight and Sibling Frames Based on Generalized Bernstein Polynomials

2016 ◽  
Vol 2016 ◽  
pp. 1-13
Author(s):  
Ting Cheng ◽  
Xiaoyuan Yang
Keyword(s):  
Bernstein Polynomials ◽  
Approximation Order ◽  
Refinable Functions ◽  
Numerical Examples ◽  
Compactly Supported ◽  
Derived Properties ◽  
Cascade Algorithms ◽  
Generalized Bernstein Polynomials

We obtain a family of refinable functions based on generalized Bernstein polynomials to provide derived properties. The convergence of cascade algorithms associated with the new masks is proved, which guarantees the existence of refinable functions. Then, we analyze the symmetry, regularity, and approximation order of the refinable functions, which are of importance. Tight and sibling frames are constructed and interorthogonality of sibling frames is demonstrated. Finally, we give numerical examples to explicitly illustrate the construction of the proposed approach.

Generalized Bernstein Polynomials

2004 ◽  
Vol 44 (1) ◽  
pp. 63-78 ◽  
Author(s):  
Stanisław Lewanowicz ◽  
Paweł Woźny
Keyword(s):  
Bernstein Polynomials ◽  
Generalized Bernstein Polynomials

scholarly journals A generalization of the Bernstein polynomials

1999 ◽  
Vol 42 (2) ◽  
pp. 403-413 ◽  
Author(s):  
Haul Oruç ◽  
George M. Phillips ◽  
Philip J. Davis
Keyword(s):  
Bernstein Polynomials ◽  
Classical Case ◽  
Geometric Progression ◽  
Generalized Bernstein Polynomials

This paper is concerned with a generalization of the classical Bernstein polynomials where the function is evaluated at intervals which are in geometric progression. It is shown that, when the function is convex, the generalized Bernstein polynomials Bn are monotonic in n, as in the classical case.

scholarly journals Convergence of Generalized Bernstein Polynomials

2002 ◽  
Vol 116 (1) ◽  
pp. 100-112 ◽  
Author(s):  
Alexander Il'inskii ◽  
Sofiya Ostrovska
Keyword(s):  
Bernstein Polynomials ◽  
Generalized Bernstein Polynomials

The r-monotonicity of generalized Bernstein polynomials

2012 ◽  
Vol 55 (3) ◽  
pp. 797-807
Author(s):  
Laiyi Zhu ◽  
Zhiyong Huang
Keyword(s):  
Continuous Function ◽  
Bernstein Polynomials ◽  
Sufficient Condition ◽  
Generalized Bernstein Polynomials

AbstractLet f ∊ C[0, 1] and let the Bn(f, q; x) be generalized Bernstein polynomials based on the q-integers that were introduced by Phillips. We prove that if f is r-monotone, then Bn(f, q; x) is r-monotone, generalizing well-known results when q = 1 and the results when r = 1 and r = 2 by Goodman et al. We also prove a sufficient condition for a continuous function to be r-monotone.

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