Exact solutions of some approximation problems by means or positive operators (Q753051)

Let T be the interval [-\(\pi\),\(\pi\) ] with identified endpoints and let C(T) be endowed with supremum norm. Let \(u\in L_ 1(T)\), \(u\geq 0\) a.e. Consider the operator \(Lx(t)=(1/\pi)\int^{\pi}_{-\pi}u(t-s)x(s)ds,\) \(x\in C(T)\), \(t\in T\). For \(n\geq 1\), the author determines \(\sup \{\| Lx-x\|:\) \(x^{(n-1)}\) is absolutely continuous on T and vrai sup\(| x^{(n)}| \leq 1\}\) and discusses previous related results.