Explicit bounds on some nonlinear retarded integral inequalities.
We construct explicitly piecewise affine mappings u:ℝ ⁿ → ℝ ⁿ with affine boundary data satisfying the constraint div u = 0. As an application of the construction we give short and direct proofs of the main approximation lemmas with constraints in convex integration theory. Our approach provides direct proofs avoiding approximation by smooth mappings and works in all dimensions n ≥ 2. After a slight modification of our construction, the constraint div u = 0 can be turned into det Du = 1, giving...
We deal with projective limits of classes of functions and prove that: (a) the Chebyshev polynomials constitute an absolute Schauder basis of the nuclear Fréchet spaces ; (b) there is no continuous linear extension map from into ; (c) under some additional assumption on , there is an explicit extension map from into by use of a modification of the Chebyshev polynomials. These results extend the corresponding ones obtained by Beaugendre in [1] and [2].
In this work, we communicate the topic of complex Lie algebroids based on the extended fractional calculus of variations in the complex plane. The complexified Euler-Lagrange geodesics and Wong's fractional equations are derived. Many interesting consequences are explored.
Letg:U→ℝ (U open in ℝn) be an analytic and K-subanalytic (i. e. definable in ℝanK, whereK, the field of exponents, is any subfield ofℝ) function. Then the set of points, denoted Σ, whereg does not admit an analytic extension is K-subanalytic andg can be extended analytically to a neighbourhood of Ū.
For a large class of Hardy fields their extensions containing non--germs are constructed. Hardy fields composed of only non--germs, apart from constants, are also considered.
It is shown that times Peano differentiable functions defined on a closed subset of and satisfying a certain condition on that set can be extended to times Peano differentiable functions defined on if and only if the th order Peano derivatives are Baire class one functions.
We are given data α₁,..., αₘ and a set of points E = x₁,...,xₘ. We address the question of conditions ensuring the existence of a function f satisfying the interpolation conditions , i = 1,...,m, that is also n-convex on a set properly containing E. We consider both one-point extensions of E, and extensions to all of ℝ. We also determine bounds on the n-convex functions satisfying the above interpolation conditions.
We investigate several extension properties of Fréchet differentiable functions defined on closed sets for o-minimal expansions of real closed fields.
Let be a closed set, a positive integer and a function defined on so that the -th Peano derivative relative to exists. The major result of this paper is that if has finite Denjoy index, then has an extension, , to which is times Peano differentiable on with on for .
The paper treats functions which are defined on closed subsets of [0,1] and which are k times Peano differentiable. A necessary and sufficient condition is given for the existence of a k times Peano differentiable extension of such a function to [0,1]. Several applications of the result are presented. In particular, functions defined on symmetric perfect sets are studied.