Polynomial Approximation Theory

2007 ◽  
pp. 267-326
Keyword(s):  
Polynomial Approximation ◽  
Approximation Theory

Inequalities and Inverse Theorems in Restricted Rational Approximation Theory

1980 ◽  
Vol 32 (2) ◽  
pp. 354-361 ◽  
Author(s):  
Peter Borwein
Keyword(s):  
Rational Function ◽  
Rational Approximation ◽  
Polynomial Approximation ◽  
Approximation Theory ◽  
Supremum Norm ◽  
Bernstein Type ◽  
Real Polynomials ◽  
Inverse Theorems ◽  
Image Position ◽  
Bernstein Type Inequalities

The following lemma, part a) due to S. N. Bernstein and part b) due to A. A. Markov, is fundamental to the proofs of many inverse theorems in polynomial approximation theory.LEMMA 1. [2, p. 62 and p. 67] Let ∏n denote the real polynomials of degree at most n. Let p ∈ ∏n, thena)b)From the lemma one can deduce, for example:THEOREM 1. Iƒ there is a sequence of polynomials pn ∈ ∏n and a δ > 0 so that ‖f – pn‖[a, b] ≧ A/nk + δ then f is k times continuously differentiate on (a, b).We shall refer to inequalities, such as those of Lemma 1 that bound the derivative r′ of a rational function of degree n in terms of its supremum norm ‖r‖[a, b] and n, as Bernstein-type inequalities.

Polynomial approximation theory for smooth functions

2009 ◽  
pp. 109-116
Author(s):  
Jan S. Hesthaven ◽  
Sigal Gottlieb ◽  
David Gottlieb
Keyword(s):  
Polynomial Approximation ◽  
Approximation Theory ◽  
Smooth Functions
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