On the Fourier sine and Kontorovich–Lebedev generalized convolution transforms and applications
UDC 517.5 We study a generalized convolutions for the Fourier sine and Kontorovich - Lebedev transforms $ (h\underset{F_s,K}\ast f)(x)$ in a two-parameter function space $L_p^{\alpha, \beta}(\Bbb R_+)$. We obtain several estimates for the norms and prove a Young-type inequality for this generalized...
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| Datum: | 2020 |
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| 1. Verfasser: | |
| Format: | Artikel |
| Sprache: | Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2020
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| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/2370 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
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Ukrains’kyi Matematychnyi Zhurnal| Zusammenfassung: | UDC 517.5
We study a generalized convolutions for the Fourier sine and Kontorovich - Lebedev transforms $ (h\underset{F_s,K}\ast f)(x)$ in a two-parameter function space $L_p^{\alpha, \beta}(\Bbb R_+)$. We obtain several estimates for the norms and prove a Young-type inequality for this generalized convolution.
We impose necessary and sufficient conditions on the kernel $h$ to ensure that the generalized convolution transform$$D_h\colon f\mapsto D_{h}[f] = \left(1-\dfrac{d^2}{dx^2}\right)(h\underset{F_s,K} \ast f)(x)$$is a unitary operator in $L_2(\Bbb R_+)$ (Watson-type theorem) and derive its inverse formula. Finally, we apply these results to an integrodifferential equation and obtain an estimate for the solution in the $L_p$-norm. |
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