On variation preserving operators

Анотація

For a piecewise-continuous function f on [0; 1] we denote by n(f) the number of its sign changes. By K_n[0; 1] we denote the set of piecewise-continuous functions f on [0; 1] such that n(f) ≤ n. We prove that for any n ≥ 2 there are no integral transforms K^̃f(x)=K(x, y)f(y)dy with a continuous kernel K(x, y) such that n(K^̃f) = n(f), for every fÎ K_n[0; 1]. We give an example of a continuous kernel K(x, y) such that n(K^̃f) = n(f), for every fÎK₁[0; 1].