Properties of Generalized Modulus of Smoothness and Generalized Modulus of Convexity
Based on the definitions of generalized modulus of smoothness, the relation between generalized modulus of smoothness and t in the Banach spaces is studied, which proves three equivalent conditions of uniform normal structure and four equivalent propositions of generalized modulus of smoothness. In...
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Harbin University of Science and Technology Publications
2020-02-01
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| Series: | Journal of Harbin University of Science and Technology |
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| Online Access: | https://hlgxb.hrbust.edu.cn/#/digest?ArticleID=1834 |
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| author | ZHAO Liang YU Yang |
| author_facet | ZHAO Liang YU Yang |
| author_sort | ZHAO Liang |
| collection | DOAJ |
| description | Based on the definitions of generalized modulus of smoothness, the relation between generalized modulus of smoothness and t in the Banach spaces is studied, which proves three equivalent conditions of uniform normal structure and four equivalent propositions of generalized modulus of smoothness. In addition, which is proved that the Banach space and the super reflexive Banach space satisfy conditions of limt→0ραX（t）t12andραX（t）<α+32tω（x）－1，tω（X）≤1 have uniform normal structure.ραX（t）andω（X）are generalized modulus of smoothness and weak orthogonal coefficient respectively. Finally, which gives an inequality about the generalized convex modulus when ‖x‖2+‖y‖2=2,x,y∈X. |
| format | Article |
| id | doaj-art-c33e20ac0244464bb4bb07d10c7245d6 |
| institution | DOAJ |
| issn | 1007-2683 |
| language | zho |
| publishDate | 2020-02-01 |
| publisher | Harbin University of Science and Technology Publications |
| record_format | Article |
| series | Journal of Harbin University of Science and Technology |
| spelling | doaj-art-c33e20ac0244464bb4bb07d10c7245d62025-08-20T03:04:49ZzhoHarbin University of Science and Technology PublicationsJournal of Harbin University of Science and Technology1007-26832020-02-01250114414810.15938/j.jhust.2020.01.022Properties of Generalized Modulus of Smoothness and Generalized Modulus of ConvexityZHAO Liang0YU Yang1School of Sciences, Harbin University of Science and Technology, Harbin 150080, ChinaSchool of Sciences, Harbin University of Science and Technology, Harbin 150080, ChinaBased on the definitions of generalized modulus of smoothness, the relation between generalized modulus of smoothness and t in the Banach spaces is studied, which proves three equivalent conditions of uniform normal structure and four equivalent propositions of generalized modulus of smoothness. In addition, which is proved that the Banach space and the super reflexive Banach space satisfy conditions of limt→0ραX（t）t12andραX（t）<α+32tω（x）－1，tω（X）≤1 have uniform normal structure.ραX（t）andω（X）are generalized modulus of smoothness and weak orthogonal coefficient respectively. Finally, which gives an inequality about the generalized convex modulus when ‖x‖2+‖y‖2=2,x,y∈X.https://hlgxb.hrbust.edu.cn/#/digest?ArticleID=1834generalized modulus of smoothnessgeneralized modulus of convexityuniformly nonsquareuniform normal structure |
| spellingShingle | ZHAO Liang YU Yang Properties of Generalized Modulus of Smoothness and Generalized Modulus of Convexity Journal of Harbin University of Science and Technology generalized modulus of smoothness generalized modulus of convexity uniformly nonsquare uniform normal structure |
| title | Properties of Generalized Modulus of Smoothness and Generalized Modulus of Convexity |
| title_full | Properties of Generalized Modulus of Smoothness and Generalized Modulus of Convexity |
| title_fullStr | Properties of Generalized Modulus of Smoothness and Generalized Modulus of Convexity |
| title_full_unstemmed | Properties of Generalized Modulus of Smoothness and Generalized Modulus of Convexity |
| title_short | Properties of Generalized Modulus of Smoothness and Generalized Modulus of Convexity |
| title_sort | properties of generalized modulus of smoothness and generalized modulus of convexity |
| topic | generalized modulus of smoothness generalized modulus of convexity uniformly nonsquare uniform normal structure |
| url | https://hlgxb.hrbust.edu.cn/#/digest?ArticleID=1834 |
| work_keys_str_mv | AT zhaoliang propertiesofgeneralizedmodulusofsmoothnessandgeneralizedmodulusofconvexity AT yuyang propertiesofgeneralizedmodulusofsmoothnessandgeneralizedmodulusofconvexity |