On the Taylor Differentiability in Spaces Lp, 0 < p ≤ ∞

The function \(f\in L_p[I], \;p>0,\) is called \((k,p)\)-differentiable at a point \(x_0\in I\) if there exists an algebraic polynomial of \(\pi\) of degree no more than \(k\) for which holds \( \Vert f-\pi \Vert_{L_p[J_h]} = o(h^{k+\frac{1}{p}}), \) where \(\;J_h=[x_0-h; x_0+h]\cap I.\) At an internal point for \(k=1\) and \(p=\infty\) this is equivalent to the usual definition of the function differentiability. At an interior point for \(k=1\) and \(p=\infty\), the definition is equivalent to the usual differentiability of the function. There is a standard "hierarchy" for the existence of differentials(if \(p_1<p_2,\) then \((k,p_2)\)-differentiability should be \((k,p_1)\)-differentiability. In the works of S.N. Bernstein, A.P. Calderon and A. Zygmund were given applications of such a construction to build a description of functional spaces (\(p=\infty\)) and the study of local properties of solutions of differential equations \((1\le p\le\infty)\), respectively. This article is related to the first mentioned work. The article introduces the concept of uniform differentiability. We say that a function \(f\), \((k,p)\)-differentiable at all points of the segment \(I\), is uniformly \((k,p)\)-differentiable on \(I\) if for any number \(\varepsilon>0\) there is a number \(\delta>0\) such that for each point \(x\in I\) runs \( \Vert f-\pi\Vert_{L_p[J_h]}<\varepsilon\cdot h^{k+\frac{1}{p}} \; \) for \(0<h<\delta, \; J_h = [x\!-\!H; x\!+\!h]\cap I,\) where \(\pi\) is the polynomial of the terms of the \((k, p)\)-differentiability at the point \(x\). Based on the methods of local approximations of functions by algebraic polynomials it is shown that a uniform \((k,p)\)-differentiability of the function \(f\) at some \(1\le p\le\infty\) implies \(f\in C^k[I].\) Therefore, in this case the differentials are "equivalent". Since every function from \(C^k[I]\) is uniformly \((k,p)\)-differentiable on the interval \(I\) at \(1\le p\le\infty,\) we obtain a certain criterion of belonging to this space. The range \(0<p<1,\) obviously, can be included into the necessary condition the membership of the function \(C^k[I]\), but the sufficiency of Taylor differentiability in this range has not yet been fully proven.

Uniform asymptotic smoothness of norms

We study a notion of smoothness of a norm on a Banach space X which generalizes the notion of uniform differentiability and is formulated in terms of unicity of Hahn Banach extensions of functionals on block subspaces of a fixed Schauder basis S in X. Variants of this notion have already been used in estimating moduli of convexity in some spaces or in fixed point theory. We show that the notion can also be used in studying the convergence of expansions coefficient of elements of X* along the dual basis S*.

Fast local convergence with single and multistep methods for nonlinear equations

The authors have noticed an oversight in Section 3.1 of this paper. To correct this error it is necessary to assume a uniform differentiability condition onG(x,h). This is required, for example, to imply δ on line 4 of p. 178 can be chosen independent ofh. For brevity we note that this oversight has been corrected in [1], which is available from the authors. Also, the results of subsequent sections are unaffected.