Estimates of probabilistic widths of the diagonal operator of finite-dimensional sets with the Gaussian measure
© Zhou and Li; licensee Springer 2013
Received: 20 October 2012
Accepted: 15 May 2013
Published: 3 June 2013
In this paper, we estimate the asymptotic orders of probabilistic and average widths of the compact embedding operators from the Sobolev space into () with the Gaussian measure.
MSC:41A10, 41A46, 42A61, 46C99.
1 Introduction and main results
Problems of n-widths in the approximation theory have by now been studied in depth. A great deal of classical problems have been solved, and interesting new developments have appeared. For example, the problems of probabilistic, average and stochastic widths, which can reflect the behavior of function on the whole class and give information about the measure of the elements in the class that can be approximated to this or that degree, are the problems of this kind. For the results related to the probabilistic, average and stochastic widths, the reader may be referred to Sul’din [1, 2], Traub et al. , Maiorov [4–7], Mathé [8–12], Sun [13, 14], and Ritter . The new developments in this direction can be found in Fang’s papers [16–21]. Moreover, Carl and Pajor  proved an inequality with respect to the Gelfand numbers of an operator u from into a Hilbert space, from which one can immediately derive the inequality related to the Kolmogorov numbers by the known duality. In this article we continue the previous works and prove the estimates of probabilistic widths of the diagonal operators from onto .
where G is any Borel subset in . Obviously, .
Denote by the ball of radius ρ in . Let .
where , .
Maiorov in  proved the following result.
Theorem A 
In , Carl and Pajor proved the following result with respect to Gelfand numbers of an operator with values in a Hilbert space.
Theorem B 
for , , where is a universal constant.
Theorem A shows the asymptotic expression of the probabilistic widths of the identity embedding from into , .
- (b)Theorem B gives the upper estimate of Gelfand numbers of operators from into a Hilbert space, and some of its striking applications in the geometry of Banach spaces and Rademacher processes can be found in . By the dual relation, it is easy to obtain the similar upper estimate of the Kolmogorov’s N-widths of operators from into , i.e.,
Motivated by Theorems A and B, in general cases, here we investigate the asymptotic estimate of probabilistic widths for diagonal operators from onto , .
Now we are in a position to formulate our main results.
2 Proof of main results
In order to prove Theorems 1 and 2, we also need some auxiliary assertions.
From (3) we obtain the assertion of the lemma by .
Next, assume that is a symmetric transformation of , then there is an orthogonal matrix U of order m such that the matrix is a diagonal matrix. Since the Gaussian measure is invariant for orthogonal transformation, the result holds for symmetric transformation .
Finally, assume that is a general invertible linear transformation from onto , then there are two matrices U and S such that , where U is an orthogonal matrix and S is a positive definite symmetric matrix. As the same reason above, the result holds for the transformation .
Thus Lemma 1 is proved. □
where is an absolute constant.
where a and are absolute constants.
where a is some absolute constant.
Consider the polyhedron . Let be the set of extremal points of Q. The set consists of vectors with k coordinates equal to and the remaining coordinates zero. This implies that , and hence .
Thus, Lemma 2 is proved. □
where is the orthogonal complement of H and . The proof of Theorem 1 is completed. □
is the deviation of K from the set in .
where the is as above, the infima are over all possible subsets of measure and all subspaces with .
Consequently, letting , we get that , which together with (10) completes the proof of Lemma 3. □
where Γ is the Euler Γ-function, and , depend only on p.
To estimate from below, we now need another auxiliary result.
where , , are non-zero eigenvalues of the operator rearranged as usual so that is non-increasing and each eigenvalue is repeated according to its multiplicity.
from which the result of Lemma 4 follows immediately. □
we have obtained a contradiction.
Next, assume that is a symmetric transformation of , then there is an orthogonal matrix U of order m such that the matrix is a diagonal matrix. Since the Lebesgue measure is invariant for orthogonal transformation, the result holds for symmetric transformation .
Finally, in the general case, is a general invertible linear transformation from onto , then there are two matrices U and S such that , where U is an orthogonal matrix and S is a positive definite symmetric matrix. As the same reason above, the result holds for the transformation .
Thus, we complete the proof of Lemma 5. □
for some constants and and N with . Lemma 6 is proved. □
for any element . On the unit sphere , we consider the subset .
We have obtained a contradiction.
Theorem 2 is a direct consequence of this. □
The authors thank the editor and the referees for their valuable suggestions to improve the quality of this paper. The present investigations was supported by the Natural Science Foundation of Inner Mongolia Province of China under Grant 2011MS0103.
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