Sampling Discretization of Integral Norms of the Hyperbolic Cross Polynomials

@article{Temlyakov2020SamplingDO,
  title={Sampling Discretization of Integral Norms of the Hyperbolic Cross Polynomials},
  author={Vladimir N. Temlyakov},
  journal={Proceedings of the Steklov Institute of Mathematics},
  year={2020},
  volume={312},
  pages={270-281}
}
  • V. Temlyakov
  • Published 12 May 2020
  • Mathematics, Computer Science
  • Proceedings of the Steklov Institute of Mathematics
Abstract The paper is devoted to discretization of integral norms of functions from a given finite-dimensional subspace. We use recent general results on sampling discretization to derive a new Marcinkiewicz type discretization theorem for the multivariate trigonometric polynomials with frequencies from the hyperbolic crosses. It is shown that recently developed techniques allow us to improve the known results in this direction. 
Sampling discretization of integral norms and its application
TLDR
This paper proves sampling discretization results under two standard kinds of assumptions – conditions on the entropy numbers and conditions in terms of the Nikol’skii-type inequalities, and applies these results to subspaces with special structure.
Sampling discretization and related problems
This survey addresses sampling discretization and its connections with other areas of mathematics. We present here known results on sampling discretization of both integral norms and the uniform norm
Approximation of functions with small mixed smoothness in the uniform norm
TLDR
The focus will be on the behavior of the best m-term trigonometric approximation as well as the decay of Kolmogorov and entropy numbers in the uniform norm, which have direct implications for the problem of sampling recovery in L2.

References

SHOWING 1-10 OF 50 REFERENCES
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.
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
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
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
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
Entropy numbers and Marcinkiewicz-type discretization theorem
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
On a norm and approximate characteristics of classes of multivariable functions
We introduce a space of quasicontinuous functions and study its approximate characteristics, i.e., ε-entropy and widths. We establish inequalities for norms of trigonometric polynomials in this
Constructive Sparse Trigonometric Approximation for Functions with Small Mixed Smoothness
This paper gives a constructive method, based on greedy algorithms, that provides for the classes of functions with small mixed smoothness the best possible in the sense of order approximation error
The Volume Estimates and Their Applications
Abstract : We prove new estimates for the entropy numbers of classes of multivariate functions with bounded mixed derivative. It is known that the investigation of these classes requires development
Observations on discretization of trigonometric polynomials with given spectrum
(here and below, C, C1, C2, . . . are different positive constants). Estimates like (3) for various finite-dimensional function spaces have found diverse applications in analysis. Systems Φ of
...
1
2
3
4
5
...