Entropy and Widths of Multiplier Operators on Two-Point Homogeneous Spaces

Author(s)

Kushpel, A. ^[1] ; Tozoni, S. A. ^[2]

Affiliation

^[1] Univ Leicester, Dept Math, Leicester LE1 7RH, Leics - England

^[2] Univ Estadual Campinas, Inst Matemat, BR-13083859 Campinas, SP - Brazil

• Document type: Journal article
• Source: CONSTRUCTIVE APPROXIMATION; v. 35, n. 2, p. 137-180, APR 2012.
• Web of Science Citations: 10

Abstract

In this article we continue the development of methods of estimating n-widths and entropy of multiplier operators begun in 1992 by A. Kushpel (Fourier Series and Their Applications, pp. 49-53, 1992; Ukr. Math. J. 45(1): 59-65, 1993). Our main aim is to give an unified treatment for a wide range of multiplier operators. on symmetric manifolds. Namely, we investigate entropy numbers and n-widths of decaying multiplier sequences of real numbers. Lambda = [lambda(k)](k=1)(infinity), |lambda(1)| >= |lambda(2)| >= ... , Lambda : L-p(M-d) -> L-q (M-d) on two-point homogeneous spaces M-d : S-d, P-d (R), P-d (C), Pd (H), P-16(Cay). In the first part of this article, general U(p)per and lower bounds are established for entropy and n-widths of multiplier operators. In the second part, different applications of these results are presented. In particular, we show that these estimates are order sharp in various important situations. For example, sharp order estimates are found for function sets with finite and infinite smoothness. We show that in the case of finite smoothness (i.e., |lambda(k)| (infinity)(k=1)|lambda(1)| >= |lambda(2)| >= ... , Lambda, k -> infinity), we have e(n)(Lambda U-p(S-d), L-q (S-d)) << d(n)(U-p(S-d), L-q (S-d)), n -> infinity, but in the case of infinite smoothness (i.e., |lambda k| (sic) e(-gamma kr), gamma > 0, 0 < r = 1, k.8), we have e(n)(Lambda U-p(S-d), Lq (Sd)) >> dn(dU(p)(S-d), Lq (Sd)), n -> 8 for different p and q, where U-p(S-d) denotes the closed unit ball of Lp(S-d). (AU)

• FAPESP's process: 03/10393-8 - Alexander Konstantinovich Kushpel | Ryerson Polytechnic University - Canada
• Grantee: Sergio Antonio Tozoni
• Support Opportunities: Research Grants - Visiting Researcher Grant - International