Exact Kolmogorov-Type Inequalities with Bounded Leading Derivative in the Case of Low Smoothness
We obtain new unimprovable Kolmogorov-type inequalities for differentiable periodic functions. In particular, we prove that, for r = 2, k = 1 or r = 3, k = 1, 2 and arbitrary q, p ∈ [1, ∞], the following unimprovable inequality holds for functions \(x \in L_\infty ^r \) : $$\left\| {x^{\left( k...
Gespeichert in:
| Datum: | 2001 |
|---|---|
| Hauptverfasser: | , , , , , |
| Format: | Artikel |
| Sprache: | Russisch Englisch |
| Veröffentlicht: |
Institute of Mathematics, NAS of Ukraine
2001
|
| Online Zugang: | https://umj.imath.kiev.ua/index.php/umj/article/view/4349 |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Institution
Ukrains’kyi Matematychnyi Zhurnal| Zusammenfassung: | We obtain new unimprovable Kolmogorov-type inequalities for differentiable periodic functions. In particular, we prove that, for r = 2, k = 1 or r = 3, k = 1, 2 and arbitrary q, p ∈ [1, ∞], the following unimprovable inequality holds for functions \(x \in L_\infty ^r \) : $$\left\| {x^{\left( k \right)} } \right\|_q \leqslant \frac{{\left\| {{\phi }_{r - k} } \right\|_q }}{{\left\| {{\phi }_r } \right\|_p^\alpha }}\left\| x \right\|_p^\alpha \left\| {x^{\left( k \right)} } \right\|_\infty ^{1 - \alpha } $$ where \(\alpha = \min \left\{ {1 - \frac{k}{r},\frac{{r - k + {1 \mathord{\left/ {\vphantom {1 q}} \right. \kern-0em} q}}}{{r + {1 \mathord{\left/ {\vphantom {1 p}} \right. \kern-0em} p}}}} \right\}\) and ϕ r is the perfect Euler spline of order r. |
|---|