arXiv subjects

R. Fry

Publications and source records attributed to R. Fry.

8 recordsLinked to original sources

arXiv2010

Real analytic approximation of Lipschitz functions on Hilbert space and other Banach spaces

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 $ε>0$, there exists a Lipschitz, real analytic function $g:X\rightarrow\mathbb{R}$ such that $|f(x)-g(x)|\leq ε$ 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.

math.FA↗

arXiv2010

Real analytic approximations which almost preserve Lipschitz constants of functions defined on the Hilbert space

Let $X$ be a separable real Hilbert space. We show that for every Lipschitz function $f:X\rightarrow\mathbb{R}$, and for every $\epsilon>0$, there exists a Lipschitz, real analytic function $g:X\rightarrow\mathbb{R}$ such that $|f(x)-g(x)|\leq \epsilon$ and $\textrm{Lip}(g)\leq \textrm{Lip}(f)+\epsilon$.

math.FA↗

arXiv2010

Approximation of functions and their derivatives by analytic maps on certain Banach spaces

Let X be a separable Banach space which admits a separating polynomial; in particular X a separable Hilbert space. Let $f:X \rightarrow R$ be bounded, Lipschitz, and $C^1$ with uniformly continuous derivative. Then for each ε>0, there exists an analytic function $g:X \rightarrow R$ with $|g-f|<ε$ and $||g'-f'||<ε$.

math.FA↗

arXiv2010

Smooth extensions of functions on separable Banach spaces

Let $X$ be a Banach space with a separable dual $X^{*}$. Let $Y\subset X$ be a closed subspace, and $f:Y\to\mathbb{R}$ a $C^{1}$-smooth function. Then we show there is a $C^{1}$ extension of $f$ to $X$.

math.FA↗

arXiv2009

Approximation of Lipschitz functions by Lipschitz, C^{p} smooth functions on weakly compactly generated Banach spaces

This note corrects a gap and improves results in an earlier paper by the first named author. More precisely, it is shown that on weakly compactly generated Banach spaces X which admit a C^{p} smooth norm, one can uniformly approximate uniformly continuous functions f:X->R by Lipschitz, C^{p} smooth functions. Moreover, there is a constant C>1 so that any L-Lipschitz function f:X->R can be uniformly approximated by CL-Lipschitz, C^{p} smooth functions. This provides a `Lipschitz version' of the classical approximation results of Godefroy, Troyanski, Whitfield and Zizler.

math.FA↗

arXiv2009

Approximation by Lipschitz, C^{p} smooth functions on weakly compactly generated Banach spaces

The following result was announced in the earlier version(s) of this paper: On weakly compactly generated Banach spaces which admit a Lipschitz, C^{p} smooth bump function, one can uniformly approximate uniformly continuous, bounded, real-valued functions by Lipschitz, C^{p} smooth functions. Unfortunately, there is a gap in the proof which renders the proof in its present form incorrect

math.FA↗

arXiv2009

Approximation by Lipschitz, analytic maps on certain Banach spaces

We show that on separable Banach spaces admitting a separating polynomial, any uniformly continuous, bounded, real-valued function can be uniformly approximated by Lipschitz, analytic maps on bounded sets.

math.FA↗

arXiv2008

Corrigendum to "Approximation by C^{p}-smooth, Lipschitz functions on Banach spaces" [J. Math. Anal. Appl., 315 (2006), 599-605]

In this erratum, we recover the results from an earlier paper of the author's which contained a gap. Specifically, we prove that if X is a Banach space with an unconditional basis and admits a C^{p}-smooth, Lipschitz bump function, and Y is a convex subset of X, then any uniformly continuous function f: Y->R can be uniformly approximated by Lipschitz, C^{p}-smooth functions K:X->R. Also, if Z is any Banach space and f:X->Z is L-Lipschitz, then the approximates K:X->Z can be chosen CL-Lipschitz and C^{p}-smooth, for some constant C depending only on X.

math.FA↗