Bernstein Power Series

1964 ◽  
Vol 16 ◽  
pp. 241-252 ◽  
Author(s):  
E. W. Cheney ◽  
A. Sharma
Keyword(s):  
Power Series ◽  
Infinite Series ◽  
Approximation Theorem ◽  
Bernstein Polynomials ◽  
Starting Point ◽  
Image Position ◽  
Weierstrass Approximation Theorem

In Bernstein's proof of the Weierstrass Approximation Theorem, the polynomialsare constructed in correspondence with a function f ∊ C [0, 1] and are shown to converge uniformly to f. These Bernstein polynomials have been the starting point of many investigations, and a number of generalizations of them have appeared. It is our purpose here to consider several generalizations in the form of infinite series and to establish some of their properties.

Volterra Series Expansion for Bilinear Systems in a Nonlinear Feedback Loop

1981 ◽  
Vol 103 (2) ◽  
pp. 84-88
Author(s):  
A. Bertuzzi ◽  
C. Bruni ◽  
A. Gandolfi ◽  
A. Germani
Keyword(s):  
Series Expansion ◽  
Existence And Uniqueness ◽  
Feedback Loop ◽  
Volterra Series ◽  
Approximation Theorem ◽  
Bilinear Systems ◽  
Nonlinear Feedback ◽  
Input Output ◽  
Distributed Bilinear Systems ◽  
Weierstrass Approximation Theorem

In this paper the class of distributed bilinear systems in a nonlinear feedback loop is considered. First of all the existence and uniqueness of solution is stated when input and state belong to L2 spaces. Then the existence of the Volterra series expansion for the input-output map is proved for inputs in any sphere around the origin; the proof is based on a new version of the Weierstrass approximation theorem.

scholarly journals On constrained uniform approximation

2002 ◽  
Vol 31 (2) ◽  
pp. 103-108
Author(s):  
M. A. Bokhari
Keyword(s):  
Uniform Approximation ◽  
Approximation Theorem ◽  
Weierstrass Approximation Theorem

The problem of uniform approximants subject to Hermite interpolatory constraints is considered with an alternate approach. The uniqueness and the convergence aspects of this problem are also discussed. Our approach is based on work of P. Kirchberger (1903) and a generalization of Weierstrass approximation theorem.

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