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ban1ka1u 2022-07-02 Answered
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?
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Answers (1)

Tanner Hamilton
Answered 2022-07-03 Author has 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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