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Funktsional. Anal. i Prilozhen., 2005, Volume 39, Issue 2, Pages 64–70 (Mi faa41)  

This article is cited in 4 scientific papers (total in 4 papers)

Brief communications

Fractional Modified Dyadic Integral and Derivative on $\mathbb{R}_+$

B. I. Golubov

Moscow Engineering Physics Institute (State University)

Abstract: For functions in the Lebesgue space $L(\mathbb{R}_+)$, 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 $f\in L(\mathbb{R}_+)$, 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 $\int_{\mathbb{R}_+} f(x) dx=0$. We prove that the linear operator $J_\alpha\colon L_{J_\alpha}\to L(\mathbb{R}_+)$ is unbounded, where $L_{J_\alpha}$ is the natural domain of $J_\alpha$. A similar statement for the operator $D^{(\alpha)}\colon L_{D^{(\alpha)}}\to L(\mathbb{R}_+)$ 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 $f\in L(\mathbb{R}_+)$ and a given point $x\in\mathbb{R}_+$. 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 $x\in\mathbb{R}_+$ of $f$.

Keywords: fractional strong dyadic derivative, fractional pointwise dyadic derivative, fractional strong dyadic integral, fractional pointwise dyadic integral, Walsh–Fourier transform, dyadic convolution

DOI: https://doi.org/10.4213/faa41

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English version:
Functional Analysis and Its Applications, 2005, 39:2, 64–70

Bibliographic databases:

UDC: 517.44
Received: 28.08.2003

Citation: B. I. Golubov, “Fractional Modified Dyadic Integral and Derivative on $\mathbb{R}_+$”, Funktsional. Anal. i Prilozhen., 39:2 (2005), 64–70; Funct. Anal. Appl., 39:2 (2005), 64–70

Citation in format AMSBIB
\Bibitem{Gol05}
\by B.~I.~Golubov
\paper Fractional Modified Dyadic Integral and Derivative on $\mathbb{R}_+$
\jour Funktsional. Anal. i Prilozhen.
\yr 2005
\vol 39
\issue 2
\pages 64--70
\mathnet{http://mi.mathnet.ru/faa41}
\crossref{https://doi.org/10.4213/faa41}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2161517}
\zmath{https://zbmath.org/?q=an:1144.42302}
\transl
\jour Funct. Anal. Appl.
\yr 2005
\vol 39
\issue 2
\pages 64--70
\crossref{https://doi.org/10.1007/s10688-005-0026-4}
\scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-23744440724}


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    This publication is cited in the following articles:
    1. S. S. Volosivets, “The modified multiplicative integral and derivative of arbitrary order on the semiaxis”, Izv. Math., 70:2 (2006), 211–231  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    2. S. V. Kozyrev, “Methods and Applications of Ultrametric and $p$-Adic Analysis: From Wavelet Theory to Biophysics”, Proc. Steklov Inst. Math., 274, suppl. 1 (2011), S1–S84  mathnet  crossref  crossref  zmath  isi  elib
    3. S. S. Volosivets, “The modified $\mathbf P$-integral and $\mathbf P$-derivative and their applications”, Sb. Math., 203:5 (2012), 613–644  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    4. S. V. Kozyrev, A. Yu. Khrennikov, V. M. Shelkovich, “$p$-Adic wavelets and their applications”, Proc. Steklov Inst. Math., 285 (2014), 157–196  mathnet  crossref  crossref  isi  elib  elib
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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