Averaged characteristics of smoothness in $L_2$ and estimations for the widths values of function classes
UDC 517.5 In the space of $2\pi$-periodic functions $L_2,$ we investigate the characteristic of smoothness $\omega^{*}_{\mathcal{M}}(f,t):=$ $(1/t) \displaystyle\int\nolimits_0^t \omega_{\mathcal{M}}(f,\tau) d \tau $ obtained as a result of averaging of the generalized modulus of continuity $\omeg...
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| Date: | 2025 |
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| Main Authors: | , , , , , |
| Format: | Article |
| Language: | Ukrainian |
| Published: |
Institute of Mathematics, NAS of Ukraine
2025
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| Online Access: | https://umj.imath.kiev.ua/index.php/umj/article/view/8794 |
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| Journal Title: | Ukrains’kyi Matematychnyi Zhurnal |
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Ukrains’kyi Matematychnyi Zhurnal| Summary: | UDC 517.5
In the space of $2\pi$-periodic functions $L_2,$ we investigate the characteristic of smoothness $\omega^{*}_{\mathcal{M}}(f,t):=$ $(1/t) \displaystyle\int\nolimits_0^t \omega_{\mathcal{M}}(f,\tau) d \tau $ obtained as a result of averaging of the generalized modulus of continuity $\omega_{\mathcal{M}}(f)$ formed by using a generalized finite-difference operator $\Delta^{\mathcal{M}}_h \colon L_2 \to L_2.$ We also study some properties of the functions $\omega_{\mathcal{M}}(f)$ and $\omega^{*}_{\mathcal{M}}(f).$ For the classes of functions $W(\omega^{*}_{\mathcal{M}}, \Phi),$ where $\Phi$ is a majorant, we determine the lower and upper estimates for the values of a series of $n$-widths and the establish the condition for $\Phi$ under which we obtain the exact values of these estimates. Several exact results are illustrated by specific examples. |
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| DOI: | 10.3842/umzh.v77i2.8794 |