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 Mat. Zametki, 2006, Volume 79, Issue 2, Pages 213–233 (Mi mz2691)

Modified Dyadic Integral and Fractional Derivative on $\mathbb R_+$

B. I. Golubov

Moscow Engineering Physics Institute (State University)

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

DOI: https://doi.org/10.4213/mzm2691

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English version:
Mathematical Notes, 2006, 79:2, 196–214

Bibliographic databases:

UDC: 517.5

Citation: B. I. Golubov, “Modified Dyadic Integral and Fractional Derivative on $\mathbb R_+$”, Mat. Zametki, 79:2 (2006), 213–233; Math. Notes, 79:2 (2006), 196–214

Citation in format AMSBIB
\Bibitem{Gol06} \by B.~I.~Golubov \paper Modified Dyadic Integral and Fractional Derivative on~$\mathbb R_+$ \jour Mat. Zametki \yr 2006 \vol 79 \issue 2 \pages 213--233 \mathnet{http://mi.mathnet.ru/mz2691} \crossref{https://doi.org/10.4213/mzm2691} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2249111} \zmath{https://zbmath.org/?q=an:1135.26007} \elib{http://elibrary.ru/item.asp?id=9218879} \transl \jour Math. Notes \yr 2006 \vol 79 \issue 2 \pages 196--214 \crossref{https://doi.org/10.1007/s11006-006-0023-9} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000235913800023} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-31844436377} 

• http://mi.mathnet.ru/eng/mz2691
• https://doi.org/10.4213/mzm2691
• http://mi.mathnet.ru/eng/mz/v79/i2/p213

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This publication is cited in the following articles:
1. S. S. Volosivets, “The modified $\mathbf P$-integral and $\mathbf P$-derivative and their applications”, Sb. Math., 203:5 (2012), 613–644
2. S. V. Kozyrev, A. Yu. Khrennikov, V. M. Shelkovich, “$p$-Adic wavelets and their applications”, Proc. Steklov Inst. Math., 285 (2014), 157–196
3. S. S. Volosivets, “Modified Bessel ${\mathbf P}$-integrals and $\mathbf P$-derivatives and their properties”, Izv. Math., 78:5 (2014), 877–901
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