Nearby Chebyshev (powered) rational approximation (Q1262490)

The problem of Chebyshev approximation on compact subset X of metric space W by (powered) generalized rational functions is, given families \(\{\Phi_ 1,...,\Phi_ n\}\), \(\{\psi_ 1,...,\psi_ n\}\) in C(W) and linearly independent on X and given \(f\in C(W)\), to find an \(n+m\) tuple \(A=(a_ 1,...,a_{n+m})\) to minimize the error norm \(\| f-R^ t(A,x)\|_ X\) subject to the constraints \(\sum^{m}_{i=1}| a_{n+i}| >0\), \(\sum^{m}_{i=1}a_{n+i}\psi_ i(x)\geq 0\) (x\(\in X)\) where \[ R^ t(A,x)=[P^ t(A,x)]^ s/[Q^ t(A,x)]^ r=[\sum^{n}_{i=1}a_ i\Phi_ i^ t(x)]^ s/[\sum^{m}_{i=1}a_{n+i}\psi_ i^ t(x)]^ r \] (r,s are fixed positive integers). It is proved that if \(\{\Phi_ 1,...,\Phi_ n\}\) is independent on X, f have an admissible best approximation on X (i.e., a mapping which can be written as a ratio with positive denominators on X) then \(A^ k\) has an accumulation point, any accumulation point is best and \(\phi_ k(f_ k,X_ k)\to \phi (f,X)\), where \(\phi_ t(f,X)=\inf \{\| f-R^ t(A,x)\|_ X\), \(Q^ t(A,x)\geq 0\), \(x\in X\), \(\sum^{m}_{i=1}| a_{n+i}| =1\}\). A similar result for weighted approximation is also given.