Fractional Modified Dyadic Integral and Derivative on ℝ+

Abstract

For functions in the Lebesgue space L(R+), a modified strong dyadic integral J(alpha) and a modified strong dyadic derivative D-(alpha) of fractional order alpha > 0 are introduced. For a given function f is an element of L(R+), criteria for the existence of these integrals and derivatives are obtained. A countable set of eigenfunctions for the operators J(alpha) and D-(alpha) is indicated. The formulas D-(alpha) (J(alpha)(f)) = f and J(alpha) (D-(alpha) (f)) = f are proved for each alpha > 0 under the condition that integral(R+) f (x) dx = 0. We prove that the linear operator J alpha : L-j alpha --> L(R+) is unbounded, where L-J alpha is the natural domain of J(alpha). A similar statement for the operator D-(alpha) : L-D(alpha) --> L(R+) is proved. A modified dyadic derivative d((alpha)) (f) (x) and a modified dyadic integral j(alpha) (f) (x) are also defined for a function f is an element of L(R+) and a given point x is an element of R+. The formulas d((alpha)) (J(alpha) (f)) (x) = f (x) and j(alpha) (D-(alpha) (f)) = f (x) are shown to be valid at each dyadic Lebesgue point x is an element of R+ of f.