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25). 2. 21) is, basically, the Tschebyshev approximation problem. Indeed, instead of asking what is the largest possible value of fTv under the constraints \bjv\ < 1, / = 1, . . , n, we might ask what is the minimum value of max, \bfv\ under the constraint that f1v= 1. The optimal solutions to these two problems can be easily obtained from each other. The second problem is equivalent to a Tschebyshev approximation problem: we can use the equation fTv = 1 to eliminate one of the design variables, thus coming to a Tschebyshev problem in the remaining variables.

2. Given the total number M of points in the finite set 7\ how should we choose these points to get the best possible quality of approximation? The goal of the subsequent series of problems is to provide some answers to these two questions. , fN\ Given a finite set T C A, let us say that T is L-dense, if there exists K < oo such that the smallest K with the latter property is denoted by KL(T}. If T is not L-dense, we set KL(T} = oo. Note that Ki(T} majorates the quality of approximating the problem (Appr(A)) by (Appr(T)), and this is the quantity we shall focus on.

Look at the convex sets X/ = {x \ \aj x — b-t\ < a*}. , UM are linearly independent and a* > 0, then for every optimal solution x* to (T) there exist k + 1 indices ofalternance—there exists a subset J C {1, . . , M} of the cardinality k + 1 such that 40 Lecture 1. Linear Programming Integration formulas and Gauss points An integration formula is a formula of the type with nonnegative weights a/. , tN and nonnegative weights a \ , . . , « # ) , one may ask how rich is the set of functions for which the formula is exact.

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