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

This article is cited in 3 scientific papers (total in 3 papers)

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
Received: 11.10.2004

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
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\paper Modified Dyadic Integral and Fractional Derivative on~$\mathbb R_+$
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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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    2. 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
    3. S. S. Volosivets, “Modified Bessel ${\mathbf P}$-integrals and $\mathbf P$-derivatives and their properties”, Izv. Math., 78:5 (2014), 877–901  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
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