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Daniel Azagra - One of the best experts on this subject based on the ideXlab platform.
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Every closed convex set is the set of minimizers of some C∞ − smooth convex function
2014Co-Authors: Daniel Azagra, Juan Ferrera, Communicated Jonathan M. BorweinAbstract:Abstract. We show that for every closed convex set C in a Separable Banach Space X there is a C∞-smooth convex function f: X − → [0,∞) so that f−1(0) = C. We also deduce some interesting consequences concerning smooth approximation of closed convex sets and continuous convex functions. It is well known that if a Separable Banach Space has a C1-smooth equivalent norm, then every closed convex C set can be regarded as the set of minimizers of a C1-smooth convex function f. One can obtain such a function f by considering the inf-convolution of the smooth norm with the indicator function of C (valued 0 on C and + ∞ elsewhere). Indeed, Separable Asplund Spaces have equivalent norms with dual LUR norms and, as shown by Asplund and Rockafellar, inf-convolutions pre-serve Fréchet smoothness when one of the convex functions is a norm whose dual is LUR (see [6], Proposition 2.3 for instance). However, the inf-convolution operation does not preserve C2 smoothness of the norm, so this procedure does not provide C2-smooth convex functions with a prescribed set of minimizers, and it seems to be an open question whether for every closed convex set C in a Separable Banac
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real analytic approximation of lipschitz functions on hilbert Space and other Banach Spaces
Journal of Functional Analysis, 2012Co-Authors: Daniel Azagra, R Fry, L KeenerAbstract:Let X be a Separable Banach Space with a separating polynomial. We show that there exists C >= 1 (depending only on X) such that for every Lipschitz function f : X -> R, and every epsilon > 0, there exists a Lipschitz, real analytic function g : X -> R such that vertical bar f (x) - g(x)vertical bar <= epsilon e and Lip(g) <= C Lip(f). This result is new even in the case when X is a Hilbert Space. Furthermore, in the Hilbertian case we also show that C can be assumed to be any number greater than I.
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real analytic approximation of lipschitz functions on hilbert Space and other Banach Spaces
arXiv: Functional Analysis, 2010Co-Authors: Daniel Azagra, R Fry, L KeenerAbstract:Let $X$ be a Separable Banach Space with a separating polynomial. We show that there exists $C\geq 1$ (depending only on $X$) such that for every Lipschitz function $f:X\rightarrow\mathbb{R}$, and every $\epsilon>0$, there exists a Lipschitz, real analytic function $g:X\rightarrow\mathbb{R}$ such that $|f(x)-g(x)|\leq \epsilon$ and $\textrm{Lip}(g)\leq C\textrm{Lip}(f)$. This result is new even in the case when $X$ is a Hilbert Space. Furthermore, in the Hilbertian case we also show that $C$ can be assumed to be any number greater than 1.
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every closed convex set is the set of minimizers of some smooth convex function
Proceedings of the American Mathematical Society, 2002Co-Authors: Daniel Azagra, Juan FerreraAbstract: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.
L Keener - One of the best experts on this subject based on the ideXlab platform.
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real analytic approximation of lipschitz functions on hilbert Space and other Banach Spaces
Journal of Functional Analysis, 2012Co-Authors: Daniel Azagra, R Fry, L KeenerAbstract:Let X be a Separable Banach Space with a separating polynomial. We show that there exists C >= 1 (depending only on X) such that for every Lipschitz function f : X -> R, and every epsilon > 0, there exists a Lipschitz, real analytic function g : X -> R such that vertical bar f (x) - g(x)vertical bar <= epsilon e and Lip(g) <= C Lip(f). This result is new even in the case when X is a Hilbert Space. Furthermore, in the Hilbertian case we also show that C can be assumed to be any number greater than I.
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real analytic approximation of lipschitz functions on hilbert Space and other Banach Spaces
arXiv: Functional Analysis, 2010Co-Authors: Daniel Azagra, R Fry, L KeenerAbstract:Let $X$ be a Separable Banach Space with a separating polynomial. We show that there exists $C\geq 1$ (depending only on $X$) such that for every Lipschitz function $f:X\rightarrow\mathbb{R}$, and every $\epsilon>0$, there exists a Lipschitz, real analytic function $g:X\rightarrow\mathbb{R}$ such that $|f(x)-g(x)|\leq \epsilon$ and $\textrm{Lip}(g)\leq C\textrm{Lip}(f)$. This result is new even in the case when $X$ is a Hilbert Space. Furthermore, in the Hilbertian case we also show that $C$ can be assumed to be any number greater than 1.
S. Ferrari - One of the best experts on this subject based on the ideXlab platform.
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sobolev Spaces with respect to a weighted gaussian measure in infinite dimensions
Infinite Dimensional Analysis Quantum Probability and Related Topics, 2019Co-Authors: S. FerrariAbstract:Let X be a Separable Banach Space endowed with a nondegenerate centered Gaussian measure μ and let w be a positive function on X such that w ∈ W1,s(X,μ) and log w ∈ W1,t(X,μ) for some s > 1 and t >...
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maximal sobolev regularity for solutions of elliptic equations in infinite dimensional Banach Spaces endowed with a weighted gaussian measure
Journal of Differential Equations, 2016Co-Authors: Gianluca Cappa, S. FerrariAbstract:Let X be a Separable Banach Space endowed with a non-degenerate centered Gaussian measure μ. The associated Cameron–Martin Space is denoted by H. Let ν=e−Uμ, where U:X→R is a sufficiently regular convex and continuous function. In this paper we are interested in the W2,2 regularity of the weak solutions of elliptic equations of the type λu−Lνu=f, where λ>0, f∈L2(X,ν) and Lν is the self-adjoint operator associated with the quadratic form (ψ,φ)↦∫X〈∇Hψ,∇Hφ〉Hdνψ,φ∈W1,2(X,ν).
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maximal sobolev regularity for solutions of elliptic equations in infinite dimensional Banach Spaces endowed with a weighted gaussian measure
arXiv: Analysis of PDEs, 2016Co-Authors: Gianluca Cappa, S. FerrariAbstract:Let $X$ be a Separable Banach Space endowed with a non-degenerate centered Gaussian measure $\mu$. The associated Cameron-Martin Space is denoted by $H$. Let $\nu=e^{-U}\mu$, where $e^{-U}$ is a sufficiently regular weight and $U:X\rightarrow\mathbb{R}$ is a convex and continuous function. In this paper we are interested in the $W^{2,2}$ regularity of the weak solutions of elliptic equations of the type \[\lambda u-L_\nu u=f,\] where $\lambda>0$, $f\in L^2(X,\nu)$ and $L_\nu$ is the self-adjoint operator associated with the quadratic form \[(\psi,\varphi)\mapsto \int_X\left\langle\nabla_H\psi,\nabla_H\varphi\right\rangle_Hd\nu\qquad\psi,\varphi\in W^{1,2}(X,\nu).\]
Abraham Rueda Zoca - One of the best experts on this subject based on the ideXlab platform.
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octahedral norms and convex combination of slices in Banach Spaces
Journal of Functional Analysis, 2014Co-Authors: J Guerrero, Gines Lopezperez, Abraham Rueda ZocaAbstract:Abstract We study the relation between octahedral norms, Daugavet property and the size of convex combinations of slices in Banach Spaces. We prove that the norm of an arbitrary Banach Space is octahedral if, and only if, every convex combination of w ⁎ -slices in the dual unit ball has diameter 2, which answers an open question. As a consequence we get that the Banach Spaces with the Daugavet property and its dual Spaces have octahedral norms. Also, we show that for every Separable Banach Space containing l 1 and for every e > 0 there is an equivalent norm so that every convex combination of w ⁎ -slices in the dual unit ball has diameter at least 2 − e .
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octahedral norms and convex combination of slices in Banach Spaces
arXiv: Functional Analysis, 2013Co-Authors: J Guerrero, Gines Lopezperez, Abraham Rueda ZocaAbstract:We study the relation between octahedral norms, Daugavet property and the size of convex combinations of slices in Banach Spaces. We prove that the norm of an arbitrary Banach Space is octahedral if, and only if, every convex combination of $w^*$-slices in the dual unit ball has diameter 2, which answer an open question. As a consequence we get that the Banach Spaces with the Daugavet property and its dual Spaces have octahedral norms. Also, we show that for every Separable Banach Space containing $\ell_1$ and for every $\varepsilon >0$ there is an equivalent norm so that every convex combination of $w^*$-slices in the dual unit ball has diameter at least $2-\varepsilon$.
Wang Zhigang - One of the best experts on this subject based on the ideXlab platform.
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some applications of principle of reduction in hilbert Space and Separable Banach Space
Journal of Mathematics, 2007Co-Authors: Wang ZhigangAbstract:In this paper, we investigate some applications of principle of reduction in Hilbert Space and Separable Banach Space. Utilizing principle of reduction and contraction principle of random series, we prove the criterion of convergence about independent random vectors in separated Banach Space, and obtain the abscissa of convergence of B-valued random Dirichlet series and the growth and the distribution of valus of random entire funcition. Some corresponding results about random Tayor series and random Dirichlet series whose coefficients are Rademacher sequences are extended to random series whose coefficients are independent and symmetric distributed.
