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 Funktsional. Anal. i Prilozhen.: Year: Volume: Issue: Page: Find

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

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 $f$.

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

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} 

• http://mi.mathnet.ru/eng/faa41
• https://doi.org/10.4213/faa41
• http://mi.mathnet.ru/eng/faa/v39/i2/p64

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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
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
3. S. S. Volosivets, “The modified $\mathbf P$-integral and $\mathbf P$-derivative and their applications”, Sb. Math., 203:5 (2012), 613–644
4. S. V. Kozyrev, A. Yu. Khrennikov, V. M. Shelkovich, “$p$-Adic wavelets and their applications”, Proc. Steklov Inst. Math., 285 (2014), 157–196
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