Polynomials of best uniform approximation to certain rational functions

1962 ◽  
Vol 4 (1) ◽  
pp. 345-349 ◽  
Author(s):  
T. J. Rivlin
Keyword(s):  
Uniform Approximation ◽  
Rational Functions ◽  
Best Uniform Approximation

Note on Best Approximation of |x|

1979 ◽  
Vol 22 (3) ◽  
pp. 363-366
Author(s):  
Colin Bennett ◽  
Karl Rudnick ◽  
Jeffrey D. Vaaler
Keyword(s):  
General Problem ◽  
Uniform Approximation ◽  
Best Approximation ◽  
Best Uniform Approximation ◽  
Linear Fractional Transformations ◽  
Symmetric Complex ◽  
Image Position ◽  
Complex Valued ◽  
Special Case

In this note the best uniform approximation on [—1,1] to the function |x| by symmetric complex valued linear fractional transformations is determined. This is a special case of the more general problem studied in [1]. Namely, for any even, real valued function f(x) on [-1,1] satsifying 0 = f ( 0 ) ≤ f (x) ≤ f (1) = 1, determine the degree of symmetric approximationand the extremal transformations U whenever they exist.

scholarly journals Concentration of the error between a function and its polynomial of best uniform approximation

1998 ◽  
Vol 41 (3) ◽  
pp. 447-463 ◽  
Author(s):  
Maurice Hasson
Keyword(s):  
Uniform Approximation ◽  
Algebraic Polynomial ◽  
General Class ◽  
Supremum Norm ◽  
Best Uniform Approximation ◽  
Class A ◽  
Image Position ◽  
Class Of Functions

Let f be a continuous real valued function defined on [−1, 1] and let En(f) denote the degree of best uniform approximation to f by algebraic polynomial of degree at most n. The supremum norm on [a, b] is denoted by ∥.∥[a, b] and the polynomial of degree n of best uniform approximation is denoted by Pn. We find a class of functions f such that there exists a fixed a ∈(−1, 1) with the following propertyfor some positive constants C and N independent of n. Moreover the sequence is optimal in the sense that if is replaced by then the above inequality need not hold no matter how small C > 0 is chosen.We also find another, more general class a functions f for whichinfinitely often.

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