Smoothness and differentiability in $L_{p}$
نویسندگان
چکیده
منابع مشابه
Smoothness beyond Differentiability
The traditional concept of smoothness concerns differentiability, and we consider a function to be smooth if it is k-times differentiable. This concept is important in many aspects of numerical analysis, and the quality of reconstructions of signals from certain given observations depend on this information. For instance, a k-times differentiable function on the unit cube [0, 1] can be approxim...
متن کاملFréchet directional differentiability and Fréchet differentiability
Zaj́ıček has recently shown that for a lower semi-continuous real-valued function on an Asplund space, the set of points where the function is Fréchet subdifferentiable but not Fréchet differentiable is first category. We introduce another variant of Fréchet differentiability, called Fréchet directional differentiability, and show that for any realvalued function on a normed linear space, the se...
متن کاملOn Polar Cones and Differentiability in Reflexive Banach Spaces
Let $X$ be a Banach space, $Csubset X$ be a closed convex set included in a well-based cone $K$, and also let $sigma_C$ be the support function which is defined on $C$. In this note, we first study the existence of a bounded base for the cone $K$, then using the obtained results, we find some geometric conditions for the set $C$, so that ${mathop{rm int}}(mathrm{dom} sigma_C) neqem...
متن کاملContinuity and Differentiability in ACL2
This case study shows how ACL2 can be used to reason about the real and complex numbers, using non-standard analysis. It describes some modifications to ACL2 that include the irrational real and complex numbers in ACL2’s numeric system. It then shows how the modified ACL2 can prove classic theorems of analysis, such as the intermediate-value and mean-value theorems.
متن کاملEffective genericity and differentiability
We prove that a real x is 1-generic if and only if every differentiable computable function has continuous derivative at x . This provides a counterpart to recent results connecting effective notions of randomness with differentiability. We also consider multiply differentiable computable functions and polynomial time computable functions. 2010 Mathematics Subject Classification 03E15; 03F60 (p...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
ژورنال
عنوان ژورنال: Studia Mathematica
سال: 1964
ISSN: 0039-3223,1730-6337
DOI: 10.4064/sm-25-1-81-91