All Questions
7 questions
2
votes
0
answers
88
views
Dependence and $L^2$ projections of functions
tl;dr: Is it possible that the best approximation to a nonnegative function of three variables with a bivariate function is no better than the best univariate function?
Let $w$ be a density on $\...
- 21
0
votes
1
answer
269
views
Determine if an integral expression is in $L^2(\mathbb{R})$
Note: This is a simplified version of the following question. I did not get a full response and realized can make it simpler to have my main interrogation answered. I decided to write it as a ...
5
votes
0
answers
471
views
A stronger Cauchy-Schwarz in infinite dimensional Hilbert spaces?
In this MSE and question and this MO question, stronger variants of the classical Cauchy-Schwarz inequality have been suggested in finite dimensional spaces. Can we find similar results for infinite ...
- 597
10
votes
1
answer
900
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
1
vote
1
answer
153
views
Optimal estimate in trace norm
Let $x,y$ be vectors of some Hilbert space of unit length.
Then we can consider the projection $P_x:=\langle \bullet, x \rangle x$ and similarly $P_y.$
Assume then that we know that $\left\lVert x-...
- 119
1
vote
1
answer
112
views
Orthogonal complement vector space
Let $X$ be a vector space contained in $H^{1}(\mathbb R^d),$ then we can study
$X^{\perp_{L^2}}:=\left\{ \xi \in L^2; \langle \xi, x \rangle_{L^2} =0 \ \forall x \in X \right\}$
and
$X^{\perp_{H^{-...
- 13
3
votes
3
answers
2k
views
Determining if a set is a Basis for l^2
For each $ n\ge 1$ Define the vectors $e_n = (e_{nk})$ where $ k\ge 1$ and $ e_{nk} = \frac{1}{k^n}$
Is this set a basis for $l^2$?
Thanks,
- 4,143