6
votes
Accepted
Determining if $\|f\|_\infty \leq C\, \|f\|_{2}^{2/3} $ holds under $f(0) = f(1) = 0$, $\|f'\|_2 \leq 1$
Counterexample: a smoothed version of the function $f$ given by the formula
$$f(x)=(h/2)^{1/2}(1-\max(0,|x/h-1|))\tag{1}\label{1}$$
with $h\downarrow0$.
Indeed, if we had
$$\|f\|_\infty\le C\,\|f\|_{2}...
- 118k
6
votes
Interpolation between two matrices so that $L^p$ norm is controlled
The constraint that $||Ax||_1 = ||x||_1$ is actually a very strong constraint. It actually implies that $A$ is a signed-permutation matrix, that is the matrix $A \in \{-1,0,1\}^{n \times n}$ and $A$ ...
- 161
2
votes
Surprising numerical coincidence while interpolating on Smolyak grid
This is no coincidence! The link between RBF and multivariate interpolation has been explored in the literature. In [1] for instance, the author shows that the limit of the RBF interpolation process ...
- 229
2
votes
Accepted
weakly separated sequences in RKHS are separated by Gleason metric
If a sequence is weakly separated, i.e. there exists a multiplier $\varphi_{ij}$ of multiplier norm at most one such that $\varphi_{ij}(\lambda_i)=\varepsilon, \varphi_{ij}(\lambda_j)=0$, then ...
2
votes
Thin-Plate-Spline understanding and solution
Not sure what should be the ultimate standard reference, but one book often referred to, Greenberg. Applications of Green's Functions in Science and Engineering 1971, 2015, in turn refers to even ...
- 281
2
votes
How is interpolation used in the proof of Lemma 4.1 in Tao's article Endpoint Strichartz Estimates?
Keel-Tao implicitly used two multilinear interpolation results.
Three end point multilinear interpolation
For general Banach spaces this is given as Exercise 3.13.5b in
Bergh, Jöran; Löfström, Jörgen, ...
- 37.6k
2
votes
Interpolation by holomorphic functions of small exponential type on a half-plane
This is not always possible under your conditions. For example, if $a_n=0$ for $n\geq 2$, then any function of exponential type $<\pi$ interpolating this sequence must be zero by Carlson's theorem,...
- 88.9k
1
vote
Abstract algebraic link between two problems involving polynomials and (generalized) Vandermonde matrices?
Well, after some more thinking I'm going to answer my own question. It was pretty much just a matter of linking all elements together.
Here are my notations, in $\mathbb R_N[X]$.
Note $\partial$ the ...
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