• Corpus ID: 210942643

Entropy numbers and Marcinkiewicz-type discretization theorem

@article{Dai2020EntropyNA,
  title={Entropy numbers and Marcinkiewicz-type discretization theorem},
  author={Feng Dai and A. V. Prymak and Alexei Shadrin and Vladimir N. Temlyakov and Sergei Yur'evich Tikhonov},
  journal={arXiv: Classical Analysis and ODEs},
  year={2020}
}
This paper studies the behavior of the entropy numbers of classes of functions with bounded integral norms from a given finite dimensional linear subspace. Upper bounds of these entropy numbers in the uniform norm are obtained and applied to establish a Marcinkiewicz type discretization theorem for integral norms of functions from a given finite dimensional subspace. 
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References

SHOWING 1-10 OF 29 REFERENCES
The Marcinkiewicz-Type Discretization Theorems
This 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
The Marcinkiewicz-type discretization theorems for the hyperbolic cross polynomials
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
Entropy numbers, s-numbers, and eigenvalue problems
Abstract We establish inequalities between entropy numbers and approximation numbers for operators acting between Banach spaces. Furthermore we derive inequalities between eigenvalues and entropy
Sampling discretization of integral norms
TLDR
A conditional theorem for all integral norms of functions from a given finite dimensional subspace is obtained, which is an extension of known results for q=1 and a new Marcinkiewicz type discretization for the multivariate trigonometric polynomials with frequencies from the hyperbolic crosses is derived.
Integral norm discretization and related problems
The problem of replacing an integral norm with respect to a given probability measure by the corresponding integral norm with respect to a discrete measure is discussed in the paper. The above
An inequality for the entropy numbers and its application
  • V. Temlyakov
  • Mathematics, Computer Science
    J. Approx. Theory
  • 2013
TLDR
An inequality for the entropy numbers in terms of nonlinear Kolmogorov widths is proved and upper bounds for the m-term approximation are obtained by using the Weak Relaxed Greedy Algorithm with respect to a system which is not a dictionary.
Interpolation and integral norms of hyperbolic polynomials
The integral norm on the subspace of multivariate trigonometric polynomials with harmonics from the “hyperbolic cross” is equivalent to the interpolation norm taken on a finite set of points whose
On the entropy numbers of the mixed smoothness function classes
  • V. Temlyakov
  • Mathematics, Computer Science
    J. Approx. Theory
  • 2017
TLDR
A new method of proving the upper bounds for the entropy numbers is developed, based on recent developments of nonlinear approximation, in particular, on greedy approximation.
Entropy numbers of diagonal operators between symmetric Banach spaces
Abstract We give the exact order of the dyadic entropy numbers of the identities from l n p to l n r where p r . Weaker estimates can be found in [3, 4]. The crucial lemma is a combinatorial result
Subspaces of small codimension of finite-dimensional Banach spaces
Given a finite-dimensional Banach space E and a Euclidean norm on E, we study relations between the norm and the Euclidean norm on subspaces of E of small codimension. Then for an operator taking
...
1
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