Gâteaux derivatives and their applications to approximation in Lorentz spaces Γp,w |
| |
| Authors: | Maciej Ciesielski Anna Kamińska Ryszard Płuciennik |
| |
| Institution: | 1. Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152, USA;2. Phone: +1 901 6782482, Fax: +1 901 6782480;3. Institute of Mathematics, Poznań University of Technology, Piotrowo 3A, 60‐965 Poznań, Poland;4. Phone: +48 616652353, Fax: +48 616652348 |
| |
| Abstract: | We establish the formulas of the left‐ and right‐hand Gâteaux derivatives in the Lorentz spaces Γp,w = {f: ∫0α (f **)p w < ∞}, where 1 ≤ p < ∞, w is a nonnegative locally integrable weight function and f ** is a maximal function of the decreasing rearrangement f * of a measurable function f on (0, α), 0 < α ≤ ∞. We also find a general form of any supporting functional for each function from Γp,w , and the necessary and sufficient conditions for which a spherical element of Γp,w is a smooth point of the unit ball in Γp,w . We show that strict convexity of the Lorentz spaces Γp,w is equivalent to 1 < p < ∞ and to the condition ∫0∞ w = ∞. Finally we apply the obtained characterizations to studies the best approximation elements for each function f ∈ Γp,w from any convex set K ? Γp,w (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim) |
| |
| Keywords: | Gâ teaux derivatives strict convexity best approximants Lorentz spaces |
|
|
