The Marcinkiewicz-Type Discretization Theorems

@article{Temlyakov2018TheMD,
  title={The Marcinkiewicz-Type Discretization Theorems},
  author={Vladimir N. Temlyakov},
  journal={Constructive Approximation},
  year={2018},
  volume={48},
  pages={337-369}
}
  • V. Temlyakov
  • Published 10 March 2017
  • Mathematics
  • Constructive Approximation
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 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, and results on the entropy numbers in the uniform norm. 
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References

SHOWING 1-10 OF 31 REFERENCES
An inequality for the entropy numbers and its application
  • V. Temlyakov
  • Mathematics, Computer Science
    J. Approx. Theory
  • 2013
Incremental Greedy Algorithm and Its Applications in Numerical Integration
TLDR
It is shown that the Incremental Algorithm provides an efficient way for deterministic construction of cubature formulas with equal weights, which give good rate of error decay for a wide variety of function classes.
On the entropy numbers of the mixed smoothness function classes
  • V. Temlyakov
  • Mathematics, Computer Science
    J. Approx. Theory
  • 2017
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
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
Rates of convex approximation in non-hilbert spaces
This paper deals with sparse approximations by means of convex combinations of elements from a predetermined “basis” subsetS of a function space. Specifically, the focus is on therate at which the
Interlacing families II: Mixed characteristic polynomials and the Kadison{Singer problem
We use the method of interlacing polynomials introduced in our previous article to prove two theorems known to imply a positive solution to the Kadison{Singer problem. The rst is Weaver’s conjecture
Hyperbolic Cross Approximation
TLDR
A survey on classical methods developed in multivariate approximation theory, which are known to work very well for moderate dimensions and which have potential for applications in really high dimensions.
Constructive sparse trigonometric approximation and other problems for functions with mixed smoothness
Our main interest in this paper is to study some approximation problems for classes of functions with mixed smoothness. We use technique, based on a combination of results from hyperbolic cross
Greedy-Type Approximation in Banach Spaces and Applications
Abstract We continue to study the efficiency of approximation and convergence of greedy-type algorithms in uniformly smooth Banach spaces. Two greedy-type approximation methods, the Weak Chebyshev
...
1
2
3
4
...