All Questions
11 questions
10
votes
2
answers
666
views
Reference request: Extensions of Wiener's Tauberian Theorem
Wiener's Tauberian Theorem says that linear combinations of translations of a function $f$ are dense in $L^1(\mathbb{R})$ if and only if the zero set of the Fourier transform of $f$ is empty. This is ...
- 710
10
votes
1
answer
899
views
Approximation of a compactly supported function by Gaussians
Let $f:\mathbb{R}\to\mathbb{R}$ be a smooth function whose support is a closed interval, e.g. $\text{supp}(f)=[a,b]$. Then $f$ can be approximated (e.g. in $L^2$) by a linear combination of Gaussian ...
- 710
5
votes
1
answer
109
views
Simultaneous near-best approximation with respect to two norms
Suppose that $M$ is a closed infinite dimensional subspace of $L_4(0,1)$ which is also a closed subspace of $L_1(0,1)$. Hence $M$ is isomorphic to $\ell_2$ as a subspace of $L_p(0,1)$ for $1\leq p\leq ...
- 4,461
5
votes
0
answers
199
views
Standard function spaces with the approximation property
A Banach space $\mathcal{X}$ is said to have the approximation property (AP) if, for every compact set $K \subset \mathcal{X}$, there is a sequence of finite rank operators $\{U_n : \mathcal{X} \to \...
- 313
2
votes
1
answer
259
views
Are Chebyshev polynomials a Schauder basis of $\mathrm{Lip}[-1,1]$?
It is known that every Lipschitz function $f \colon [-1,1] \to \mathbb R$ can be expressed as a series in the Chebyshev polynomials $$f = \sum_{n = 0}^\infty a_n T_n $$ which is absolutely convergent ...
- 319
2
votes
1
answer
133
views
Control on dimension of image
Let $f:E\rightarrow F$ be a map between Banach spaces E and F; E finite dimensional (>0) and F infinite dimensional. Let $F$ be equipped with its weak topology and suppose that $f$ is strong-weak ...
- 5,405
2
votes
1
answer
5k
views
Smooth Approximation of Indicator Function of Convex Sets in $\mathbb{R}^n$
Let $( \mathbb{R}^n, \| \cdot \|_P)$ be the $n$-dimensional Euclidean space equipped with $\ell_p$-norm $\| \cdot \|_p$ for some $p\in [1, + \infty]$. Let $A$ be a convex set in $\mathbb{R}^n$ and ...
- 1,127
1
vote
1
answer
114
views
Example of a nonconvex Chebyshev set in a metric space with continuous projection?
Question: Is there an example of a nonconvex Chebyshev set $S$ in a metric space $(X,d)$ whose projection map is continuous?
For convexity to be well-defined, we need to assume that $X$ is a vector ...
- 710
1
vote
1
answer
259
views
Finding the set of best approximation
Given $X$=$l^1$ and its dual space $X^*=l^\infty$. Now take $f=(1, 1/2, 2/3, 3/4,...) \in X^*$. Then clearly $\|f\|_\infty = 1$. I have found that $H=\ker f$ is a proximinal hyperplane in $X$.
Note: A ...
- 85
1
vote
0
answers
174
views
Estimates of entropy of functional spaces
Let $M^n$ be a compact $n$-dimensional manifold. For $k\geq 0$ let us denote by $C^k(M)$ the Banach space of $k$ times continuously differentiable functions, and $B_{C^k}$ denote the unit ball of it.
...
- 21.8k
0
votes
1
answer
407
views
Criteria for $\epsilon$-Density
Let $Y$ be a compact, separable metric space and $X=C(Y)$ Banach space. There are many criteria for a linear subspace $Z\subseteq X$ to be dense; notably the Stone-Weierstraß theorem.
Are there ...
- 5,405