Given a multivariate rational function p ( <mrow class="MJX-TeXAtom-ORD"> <mover>

ban1ka1u

ban1ka1u

Answered question

2022-07-02

Given a multivariate rational function p ( x ) = f ( x ) g ( x ) over [ 0 , 1 ] n with p ( x ) [ 0 , 1 ], how can we come up with a polynomial approximation of p, say q ( x ) such that | p ( x ) q ( x ) | ϵ for all x ?
For the univariate case, we might use chebyshev approximation, however, what are the results for the multivariate case?

Answer & Explanation

Tanner Hamilton

Tanner Hamilton

Beginner2022-07-03Added 12 answers

Approximation of continuous functions of several variables on [ 0 , 1 ] n can be done by means of multivariate Chebyshev polynomials that are the tensor product of Chebyshev polynomials in one variable. For f C ( [ 0 , 1 ] n ),
f ( x 1 , , x n ) 1 k i N a k 1 , , k n T k 1 ( x 1 ) T k n ( x n ) .

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