Azagra Rueda, Daniel

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Azagra Rueda
Universidad Complutense de Madrid
Faculty / Institute
Ciencias Matemáticas
Análisis Matemático Matemática Aplicada
Análisis Matemático
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Now showing 1 - 10 of 46
  • Publication
    Riemann integrability and Lebesgue measurability of the composite function
    (Academic Press, 2009-06) Azagra Rueda, Daniel; Muñoz-Fernández, Gustavo A.; Seoane-Sepúlveda, Juan B.; Sánchez de los Reyes, Víctor Manuel
    If f is continuous on the interval [a, b], g is Riemann integrable (resp. Lebesgue measurable) on the interval [alpha, beta] and g([alpha, beta]) subset of [a, b], then f o g is Riemann integrable (resp. measurable) on [alpha, beta]. A well-known fact, on the other hand, states that f o g might not be Riemann integrable (resp. measurable) when f is Riemann integrable (resp. measurable) and g is continuous. If c stands for the continuum, in this paper we construct a 2(c)-dimensional space V and a c-dimensional space W of, respectively, Riemann integrable functions and continuous functions such that, for every f is an element of V \ {0} and g is an element of W \ {0} . f o g is not Riemann integrable, showing that nice properties (such as continuity or Riemann integrability) can be lost, in a linear fashion, via the composite function. Similarly we construct a c-dimensional space W of continuous functions such that for every g is an element of W \ {0} there exists a c-dimensional space V of measurable functions such that f o g is not measurable for all f is an element of V \ {0}.
  • Publication
    Rolle’s Theorem and Negligibility of Points in Infinite Dimensional Banach Spaces
    (Elsevier, 1997-09-15) Azagra Rueda, Daniel; Gómez Gil, Javier; Jaramillo Aguado, Jesús Ángel
    In this note we prove that if a differentiable function oscillates between y« and « on the boundary of the unit ball then there exists a point in the interior of the ball in which the differential of the function has norm equal or less than« . This kind of approximate Rolle’s theorem is interesting because an exact Rolle’s theorem does not hold in many infinite dimensional Banach spaces. A characterization of those spaces in which Rolle’s theorem does not hold is given within a large class of Banach spaces. This question is closely related to the existence of C1 diffeomorphisms between a Banach space X and X _ _04 which are the identity out of a ball, and we prove that such diffeomorphisms exist for every C1 smooth Banach space which can be linearly injected into a Banach space whose dual norm is locally uniformly rotund (LUR).
  • Publication
    Smooth extensions of functions on separable Banach spaces
    (Springer, 2010) Azagra Rueda, Daniel; Fry, Robb; Keener, L.
    Let X be a Banach space with a separable dual X*. Let Y subset of X be a closed subspace, and f : Y -> R a C(1)-smooth function. Then we show there is a C(1) extension of f to X.
  • Publication
    On the topological classification of starlike bodies in Banach spaces
    (Elsevier Science, 2003-08-15) Azagra Rueda, Daniel; Dobrowolski, Tadeusz
    Starlike bodies are interesting in nonlinear analysis because they are strongly related to polynomials and smooth bump functions, and their topological and geometrical properties are therefore worth studying. In this note we consider the question as to what extent the known results on topological classification of convex bodies can be generalized for the class of starlike bodies, and we obtain two main results in this line, one which follows the traditional Bessaga-Klee scheme for the classification of convex bodies (and which in this new setting happens to be valid only for starlike bodies whose characteristic cones are convex), and another one which uses a new classification scheme in terms of the homotopy type of the boundaries of the starlike bodies (and which holds in full generality provided the Banach space is infinite-dimensional).
  • Publication
    Regularization by sup-inf convolutions on Riemannian manifolds: An extension of Lasry-Lions theorem to manifolds of bounded curvature
    (Academic Press, 2015-03-15) Azagra Rueda, Daniel; Ferrera Cuesta, Juan
    We show how Lasry-Lions's result on regularization of functions defined on R-n or on Hilbert spaces by sup inf convolutions with squares of distances can be extended to (finite or infinite dimensional) Riemannian manifolds M of bounded sectional curvature. More specifically, among other things we show that if the sectional curvature K of M satisfies -K-0 <= K <= K-0 on M for some K-0 >= 0, and if the injectivity and convexity radii of M are strictly positive, then every bounded, uniformly continuous function f : M -> R can be uniformly approximated by globally C-1,C-1 functions defined by (f(lambda))(mu) = sup(z is an element of M nu is an element of M)inf {f(y) + 1/2 lambda d(z, y)(2) - 1/2 mu d(x, z)(2)} as lambda, mu -> 0(+), with 0 < mu < lambda/2. Our definition of (global) C-1,C-1 smoothness is intrinsic and natural, and it reduces to the usual one in flat spaces, but we warn the reader that, in the noncompact case, this definition differs from other notions of (rather local) C-1,C-1 smoothness that have been recently used by other authors (based on charts). The importance of this regularization method lies (rather than on the degree of smoothness obtained) on the fact that the correspondence f bar right arrow (f(lambda))(mu) is explicit and preserves many significant geometrical properties that the given functions f may have, such as invariance by a set of isometries, infima, sets of minimizers, ordering, local or global Lipschitzness, and (only when one additionally assumes that K <= 0) local or global convexity. We also give two examples showing that this result completely fails, even for (nonflat) Cartan-Hadamard manifolds, whenever f or K are not bounded.
  • Publication
    Whitney extension theorems for convex functions of the classes C1 and C1ω
    (Oxford University Press (OUP), 2017) Azagra Rueda, Daniel; Mudarra, C.
    Let C be a subset of ℝn (not necessarily convex), f : C → R be a function and G : C → ℝn be a uniformly continuous function, with modulus of continuity ω. We provide a necessary and sufficient condition on f, G for the existence of a convex function F ∈ CC1ω(ℝn) such that F = f on C and ∇F = G on C, with a good control of the modulus of continuity of ∇F in terms of that of G. On the other hand, assuming that C is compact, we also solve a similar problem for the class of C1 convex functions on ℝn, with a good control of the Lipschitz constants of the extensions (namely, Lip(F) ≲ ∥G∥∞). Finally, we give a geometrical application concerning interpolation of compact subsets K of ℝn by boundaries of C1 or C1,1 convex bodies with prescribed outer normals on K.
  • Publication
    A maximum principle for evolution Hamilton-Jacobi equations on Riemannian manifolds
    (Elsevier, 2006-11-01) Azagra Rueda, Daniel; Ferrera Cuesta, Juan; López-Mesas Colomina, Fernando
    We establish a maximum principle for viscosity subsolutions and supersolutions of equations of the form u(t) + F(t, d(x)u) = 0, u(0, x) = u(0)(x), where u(0): M -> R is a bounded uniformly continuous function, M is a Riemannian manifold, and F: [0, infinity) x T*M -> R. This yields uniqueness of the viscosity solutions of such Hamilton-Jacobi equations.
  • Publication
    Every closed convex set is the set of minimizers of some C1-smooth convex function
    (America Mathematical Society, 2002) Azagra Rueda, Daniel; Ferrera Cuesta, Juan
    The authors show that for every closed convex set C in a separable Banach space there is a nonnegative C1 convex function f such that C = {x: f(x) = 0}. The key is to show this for a closed halfspace. This result has several attractive consequences. For example, it provides an easy proof that every closed convex set is the Hausdorff limit of infinitely smooth convex bodies (Cn := {x: f(x) _ 1/n}) and that every continuous convex function is the Mosco limit of C1 convex functions.
  • Publication
    A Second Order Smooth Variational Principle on Riemannian Manifolds
    (Canadian Mathematical Society, 2010) Azagra Rueda, Daniel; Fry, Robb
    We establish a second order smooth variational principle valid for functions defined on (possibly infinite- dimensional) Riemannian manifolds which are uniformly locally convex and have a strictly positive injectivity radius and bounded sectional curvature.
  • Publication
    Uniform approximation of continuous mappings by smooth mappings with no critical points on Hilbert manifolds
    (DUKE UNIV PRESS, 2004) Azagra Rueda, Daniel; Cepelledo Boiso, M.C.
    We prove that every continuous mapping from a separable infinite-dimensional Hilbert space X into R-m can be uniformly approximated by C-infinity-smooth mappings with no critical points. This kind of result can be regarded as a sort of strong approximate version of the Morse-Sard theorem. Some consequences of the main theorem are as follows. Every two disjoint closed subsets of X can be separated by a one-codimensional smooth manifold that is a level set of a smooth function with no critical points. In particular, every closed set in X can be uniformly approximated by open sets whose boundaries are C-infinity-smooth one-codimensional submanifolds of X. Finally, since every Hilbert manifold is diffeomorphic to an open subset of the Hilbert space, all of these results still hold if one replaces the Hilbert space X with any smooth manifold M modeled on X.