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 Mat. Sb., 2007, Volume 198, Number 2, Pages 67–90 (Mi msb3780)

B. I. Golubov

Moscow Engineering Physics Institute (State University)

Abstract: On the basis of the concept of pointwise dyadic derivative dyadic distributions are introduced as continuous linear functionals on the linear space $D_d(\mathbb R_+)$ of infinitely differentiable functions compactly supported by the positive half-axis $\mathbb R_+$ together with all dyadic derivatives. The completeness of the space $D'_d(\mathbb R_+)$ of dyadic distributions is established. It is shown that a locally integrable function on $\mathbb R_+$ generates a dyadic distribution.
In addition, the space $S_d(\mathbb R_+)$ of infinitely dyadically differentiable functions on $\mathbb R_+$ rapidly decreasing in the neighbourhood of $+\infty$ is defined. The space $S'_d(\mathbb R_+)$ of dyadic distributions of slow growth is introduced as the space of continuous linear functionals on $S_d(\mathbb R_+)$. The completeness of the space $S'_d(\mathbb R_+)$ is established; it is proved that each integrable function on $\mathbb R_+$ with polynomial growth at $+\infty$ generates a dyadic distribution of slow growth.
Bibliography: 25 titles.

DOI: https://doi.org/10.4213/sm3780

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English version:
Sbornik: Mathematics, 2007, 198:2, 207–230

Bibliographic databases:

UDC: 517.982.4
MSC: 46F05, 42C10

Citation: B. I. Golubov, “Dyadic distributions”, Mat. Sb., 198:2 (2007), 67–90; Sb. Math., 198:2 (2007), 207–230

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/msb3780
• https://doi.org/10.4213/sm3780
• http://mi.mathnet.ru/eng/msb/v198/i2/p67

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This publication is cited in the following articles:
1. S. S. Volosivets, “Applications of $\mathbf P$-adic generalized functions and approximations by a system of $\mathbf P$-adic translations of a function”, Siberian Math. J., 50:1 (2009), 1–13
2. S. S. Platonov, “An Analog of Titchmarsh's Theorem for the Fourier–Walsh Transform”, Math. Notes, 103:1 (2018), 96–103
3. S. S. Platonov, “On the Fourier–Walsh Transform of Functions from Dyadic Dini–Lipschitz Classes on the Semiaxis”, Math. Notes, 108:2 (2020), 229–242
4. M. A. Karapetyants, V. Yu. Protasov, “Spaces of Dyadic Distributions”, Funct. Anal. Appl., 54:4 (2020), 272–277
5. A. Yu. Trynin, “O skhodimosti obobschenii sink-approksimatsii na klasse Privalova–Chanturiya”, Sib. zhurn. industr. matem., 24:3 (2021), 122–137
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