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

B. I. Golubov

Funkcionalʹnyj analiz i ego priloženiâ, Tome 39 (2005) no. 2, pp. 64-70.

Résumé

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 $f$.

Détail

Keywords: fractional strong dyadic derivative, fractional pointwise dyadic derivative, fractional strong dyadic integral, fractional pointwise dyadic integral, Walsh–Fourier transform
Mots-clés : dyadic convolution.

Bibliographie

[1] Butzer P. L., Wagner H. J., Appl. Analysis, 3:1 (1973), 29–46 | DOI | MR | Zbl

[2] Butzer P. L., Wagner H. J., Proc. Symp. Naval Res. Laboratory (Washington, D.C., April 18–20, 1973), 75–81 | Zbl

[3] Wagner J. H., Pros. Symp. Theory and applications of Walsh functions, Hatfield Polytechnic, UK, 1975, 101–129

[4] Butzer P. L., Wagner H. J., Analysis Math., 1 (1975), 171–196 | DOI | MR | Zbl

[5] Onneweer C. W., Analysis Math., 3 (1977), 119–130 | DOI | MR | Zbl

[6] Pal J., Simon P., Acta Math. Acad. Sci. Hung., 29 (1977), 155–164 | DOI | MR | Zbl

[7] Onneweer C. W., Appl. Analysis, 9 (1979), 267–278 | DOI | MR | Zbl

[8] Pal J., Annales Univ. Sci. Budapest. Sect. Math., 18 (1975), 49–54 | MR

[9] Golubov B. I., Matem. sb., 193:4 (2002), 37–60 | DOI | MR | Zbl

[10] Fine N. J., Trans. Amer. Math. Soc., 69 (1950), 66–77 | DOI | MR | Zbl

[11] Schipp F., Wade W. R., Simon P., Walsh series. An introduction to dyadic harmonic analysis, Akademiai Kiado, Budapest, 1990 | MR

[12] Golubov B. I., Efimov A. V., Skvortsov V. A., Ryady i preobrazovaniya Uolsha. Teoriya i primeneniya, Nauka, M., 1987 | MR | Zbl

[13] Onneweer C. W., Trans. Amer. Math. Soc., 258 (1980), 923–931 | DOI | MR

[14] Golubov B. I., Izv. RAN, Ser. matem., 67:1 (2003), 33–58 | DOI | MR | Zbl

[15] Golubov B. I., Izv. RAN, Ser. matem., 65:3 (2001), 3–14 | DOI | MR | Zbl

[16] Pal J., Schipp F., Annales Univ. Sci. Budapest. Sect. Computatorica, 8 (1987), 91–108 | MR | Zbl

[17] Schipp F., Mathematica Balkanica, 4 (1974), 541–546 | MR | Zbl

[18] Wagner J. H., Ein Differential- und Integralkalkül in der Walsh–Fourier Analysis mit Anwendungen, Westdeutscher Verlag, Köln–Opladen, 1974 | MR

[19] Pal J., Schipp F., Theory and applications of Gibbs derivatives, Proc. First Intern. Workshop on Gibbs Derivatives, Math. Inst., Beograd, 1989, 103–113