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Mathematical Notes

, Volume 79, Issue 1–2, pp 196–214 | Cite as

Modified Dyadic Integral and Fractional Derivative on ℝ+

  • B. I. Golubov
Article

Abstract

For functions from the Lebesgue space L(ℝ+), we introduce the modified strong dyadic integral Jα and the fractional derivative D(α) of order α > 0. We establish criteria for their existence for a given function fL(ℝ+). We find a countable set of eigenfunctions of the operators D(α) and Jα, α > 0. We also prove the relations D(α)(Jα(f)) = f and Jα(D(α)(f)) = f under the condition that \(\smallint _{\mathbb{R}_ + } f(x)dx = 0\). We show the unboundedness of the linear operator \(J_\alpha :L_{J_{_\alpha } } \to L(\mathbb{R}_ + )\), where L J α is its natural domain of definition. A similar assertion is proved for the operator \(D^{(\alpha )} :L_{D^{(\alpha )} } \to L(\mathbb{R}_ + )\). Moreover, for a function fL(ℝ+) and a given point x ∈ ℝ+, we introduce the modified dyadic derivative d(α)(f)(x) and the modified dyadic integral jα(f)(x). We prove the relations d(α)(Jα(f))(x) = f(x) and jα(D(α)(f)) = f(x) at each dyadic Lebesgue point of the function f.

Key words

fractional derivative and integral strong dyadic integral and derivative dyadic Lebesgue point Fourier-Walsh transform generalized Walsh functions dyadic group 

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Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • B. I. Golubov
    • 1
  1. 1.Moscow Institute of Engineering Physics (State University)MoscowRussia

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