A modified strong dyadic integral and derivative

For a function f is an element of L(R+) its modified strong dyadic integral J(f) and the modified strong dyadic derivative D(f) are defined. A criterion for the existence of a modified strong dyadic integral for an integrable function is proved, and the equalities J(D(f)) = f and D(J(f)) = f are established under the assumption that integral(R+) f(x) dx = 0. A countable system of eigenfunctions of the operators D and J is found. The linear span L of the set is shown to be dense in the dyadic Hardy space H(R+), and the linear operator (J) over tilde: L --> L(R+), (J) over tilde (f) = J(f)(similar to), is proved to be bounded. Hence this operator can be uniquely continuously extended to H(R+) and the resulting linear operator (J) over tilde: H(R+) --> L(R+) is bounded.