On Modified Strong Dyadic Integral and Derivative
For functions f ∈ L(R +), we define a modified strong dyadic integral J(f) ∈ L(R +) and a modified strong dyadic derivative D(f) ∈ L(R +). We establish a necessary and sufficient condition for the existence of the modified strong dyadic integral J(f). Under the condition \(\smallint _{R_ + }\) f...
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| Date: | 2002 |
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| Main Authors: | , |
| Format: | Article |
| Language: | Russian English |
| Published: |
Institute of Mathematics, NAS of Ukraine
2002
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/4101 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
| Download file: |  |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | For functions f ∈ L(R +), we define a modified strong dyadic integral J(f) ∈ L(R +) and a modified strong dyadic derivative D(f) ∈ L(R +). We establish a necessary and sufficient condition for the existence of the modified strong dyadic integral J(f). Under the condition \(\smallint _{R_ + }\) f(x)dx = 0, we prove the equalities J(D(f)) = f and D(J(f)) = f. We find a countable set of eigenfunctions of the operators J and D. We prove that the linear span L of this set is dense in the dyadic Hardy space H(R +). For the functions f ∈ H(R +), we define a modified uniform dyadic integral J(f) ∈ L ∞(R +). |
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