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Equivalence of some integrals and modulus of smoothness
| Content Provider | Semantic Scholar |
|---|---|
| Author | Zhu, Guoliang |
| Copyright Year | 2009 |
| Abstract | Abstract Let f ∈ C ( R ) . We are interested in lower and upper bounds of the integrals ∫ h H Δ t 2 f ( x ) t 1 + α d t , where 0 h H ⩽ ∞ and Δ t 2 f ( x ) = f ( x + t ) + f ( x − t ) − 2 f ( x ) . We will show that for fixed α > 3 there holds h α ‖ ∫ h ∞ Δ t 2 f ( ⋅ ) t 1 + α d t ‖ ≍ ω 2 ( f , h ) , ∀ f ∈ C ( R ) , h > 0 , where ω 2 ( f , h ) is the modulus of smoothness defined by ω 2 ( f , t ) = sup 0 δ ⩽ t ‖ Δ δ 2 f ‖ . For α > 2 we will show that for some 1 ⩽ A ∞ the following relations are valid: h α sup H > h ‖ ∫ h H Δ t 2 f ( ⋅ ) t 1 + α d t ‖ ≍ h α max h ⩽ t ⩽ A h ‖ ∫ t ∞ Δ u 2 f ( ⋅ ) u 1 + α d u ‖ ≍ ω 2 ( f , h ) , ∀ f ∈ C ( R ) , h > 0 . |
| Starting Page | 793 |
| Ending Page | 799 |
| Page Count | 7 |
| File Format | PDF HTM / HTML |
| DOI | 10.1016/j.jmaa.2009.03.054 |
| Volume Number | 356 |
| Alternate Webpage(s) | https://core.ac.uk/download/pdf/82058414.pdf |
| Language | English |
| Access Restriction | Open |
| Content Type | Text |
| Resource Type | Article |
