Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userMarcinkiewicz-type discretization of $L^p$-norms under the Nikolskii-type inequality assumption
132
0
0.0
(
0
)
Authors
Egor Kosov
Ask ChatGPT about the research
No Arabic abstract
The paper studies the sampling discretization problem for integral norms on subspaces of $L^p(mu)$. Several close to optimal results are obtained on subspaces for which certain Nikolskii-type inequality is valid. The problem of norms discretization is connected with the probabilistic question about the approximation with high probability of marginals of a high dimensional random vector by sampling. As a byproduct of our approach we refine the result of O. Gu$acute{e}$don and M. Rudelson concerning the approximation of marginals. In particular, the obtained improvement recovers a theorem of J. Bourgain, J. Lindenstrauss, and V. Milman concerning embeddings of finite dimensional subspaces of $L^p[0, 1]$ into $ell_p^m$. The proofs in the paper use the recent developments of the chaining technique by R. van Handel.
rate research
Read More
The paper is devoted to discretization of integral norms of functions from a given finite dimensional subspace. This problem is very important in applications but there is no systematic study of it. We present here a new technique, which works well for discretization of the integral norm. It is a combination of probabilistic technique, based on chaining, with results on the entropy numbers in the uniform norm.
In 2006 Carbery raised a question about an improvement on the naive norm inequality $|f+g|_p^p leq 2^{p-1}(|f|_p^p + |g|_p^p)$ for two functions in $L^p$ of any measure space. When $f=g$ this is an equality, but when the supports of $f$ and $g$ are disjoint the factor $2^{p-1}$ is not needed. Carberys question concerns a proposed interpolation between the two situations for $p>2$. The interpolation parameter measuring the overlap is $|fg|_{p/2}$. We prove an inequality of this type that is stronger than the one Carbery proposed. Moreover, our stronger inequalities are valid for all $p$.
The main goal of this paper is to study the discretization problem for the hyperbolic cross trigonometric polynomials. This important problem turns out to be very difficult. In this paper we begin a systematic study of this problem and demonstrate two different techniques -- the probabilistic and the number theoretical techniques.
We revisit and comment on the Harnack type determinantal inequality for contractive matrices obtained by Tung in the nineteen sixtieth and give an extension of the inequality involving multiple positive semidefinite matrices.
We describe the Lorentz space $L(p, r), 0 < r < p, p > 1$, in terms of Orlicz type classes of functions L . As a consequence of this result it follows that Steins characterization of the real functions on $R^n$ that are differentiable at almost all the points in $R^n$, is equivalent to the earlier characterization of those functions given by A. P. Calderon.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
