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 Λ on symmetric manifolds. Namely, we investigate entropy numbers and n-widths of decaying multiplier sequences of real numbers \(\varLambda=\{\lambda_{k}\}_{k=1}^{\infty}\), |λ 1|≥|λ 2|≥⋯, \(\varLambda:L_{p}(\mathbb{M}^{d}) \rightarrow L_{q}(\mathbb{M}^{d})\) on two-point homogeneous spaces \(\mathbb{M}^{d}\): \(\mathbb{S}^{d}\), ℙd(ℝ), ℙd(ℂ), ℙd(ℍ), ℙ16(Cay). In the first part of this article, general upper 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., |λ k |≍k −γ(lnk)−ζ, γ/d>1, ζ≥0, k→∞), we have \(e_{n}(\varLambda U_{p}(\mathbb{S}^{d}), L_{q}(\mathbb{S}^{d})) \ll d_{n}(\varLambda U_{p}(\mathbb{S}^{d}), L_{q}(\mathbb{S}^{d}))\), n→∞, but in the case of infinite smoothness (i.e., \(|\lambda_{k}| \asymp e^{-\gamma k^{r}}\), γ>0, 0<r≤1, k→∞), we have \(e_{n}(\varLambda U_{p}(\mathbb{S}^{d}), L_{q}(\mathbb{S}^{d})) \gg d_{n}(\varLambda U_{p}(\mathbb{S}^{d}), L_{q}(\mathbb{S}^{d}))\), n→∞ for different p and q, where \(U_{p}(\mathbb{S}^{d})\) denotes the closed unit ball of \(L_{p}(\mathbb{S}^{d})\).
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Birman, M.S., Solomyak, M.Z.: Piecewise polynomial approximations of functions of classes \(W^{\alpha}_{p}\). Mat. Sb. (N.S.) 73(115), 331–355 (1967)
Bordin, B., Kushpel, A., Levesley, J., Tozoni, S.: n-Widths of multiplier operators on two-point homogeneous spaces. In: Chui, C., Schumaker, L.L. (eds.) Approximation Theory IX, vol. 1, Theoretical Aspects, pp. 23–30. Vanderbilt University Press, Nashville (1998)
Bordin, B., Kushpel, A., Levesley, J., Tozoni, S.: Estimates of n-widths of Sobolev’s classes on compact globally symmetric spaces of rank 1. J. Funct. Anal. 202, 307–326 (2003)
Bourgain, J., Lindenstrauss, J., Milman, V.: Approximation of zonoids by zonotopes. Acta Math. 162, 73–141 (1989)
Carl, B.: Entropy numbers, s-numbers and eigenvalue problems. J. Funct. Anal. 41(3), 290–306 (1981)
Cartan, E.: Sur la determination d’un systeme orthogonal complet dans un espace de Riemann symetrique clos. Rend. Circ. Mat. Palermo 53, 217–252 (1929)
Edmunds, D.E., Triebel, H.: Entropy numbers and approximation numbers in function spaces. Proc. Lond. Math. Soc. 58(1), 137–152 (1989)
Edmunds, D.E., Triebel, H.: Function Spaces, Entropy Numbers, Differential Operators. Cambridge University Press, Cambridge (1996)
Figiel, T., Lindenstrauss, J., Milman, V.D.: The dimension of almost spherical sections of convex bodies. Acta Math. 139(1), 53–94 (1977)
Gangolli, R.: Positive definite kernels on homogeneous spaces and certain stochastic processes related to Lévy’s Browian motion of several parameters. Ann. Inst. Henri Poincaré, Sect. B (N.S.) 3, 121–226 (1967)
Helgason, S.: The Radon transform on Euclidean spaces, compact two-point homogeneous spaces and Grassmann manifolds. Acta Math. 113, 153–180 (1965)
Helgason, S.: Differential Geometry and Symmetric Spaces. Academic Press, New York (1962)
Kahane, J.P.: Some Random Series of Functions. Heath Math. Monographs. Heath, Lexington (1968)
Kashin, B.S., Tzafriri, L.A.: Lower bound for the maximum of a stochastic process. Math. Notes - Ross. Akad., 56(6), 1306–1308 (1994)
Koornwinder, T.: The addition formula for Jacobi polynomials and spherical harmonics. SIAM J. Appl. Math. 25(2), 236–246 (1973)
Kushpel, A.: On an estimate of Levy means and medians of some distributions on a sphere. In: Fourier Series and Their Applications, pp. 49–53. Inst. of Math., Kiev (1992)
Kushpel, A.: Estimates of Bernstein’s widths and their analogs. Ukr. Math. J. 45(1), 59–65 (1993)
Kushpel, A., Levesley, J., Wilderotter, K.: On the asymptotically optimal rate of approximation of multiplier operators from L p into L q . Constr. Approx. 14(2), 169–185 (1998)
Kushpel, A.K.: Estimates of entropy numbers of multiplier operators with slowly decaying coefficients. In: 48th Seminário Brasileiro de Análise, Petrópolis, RJ, pp. 711–722 (1998)
Kushpel, A.: Levy means associated with two-point homogeneous spaces and applications. In: 49th Seminário Brasileiro de Análise, Campinas, SP, pp. 807–823 (1999)
Kushpel, A.: Estimates of n-widths and ϵ-entropy of Sobolev’s sets on compact globally symmetric spaces of rank 1. In: 50th Seminário Brasileiro de Análise, São Paulo, SP, pp. 53–66 (1999)
Kushpel, A.: n-Widths of Sobolev’s classes on compact globally symmetric spaces of rank 1. In: Kopotun, K., Lyche, T., Neamtu, M. (eds.) Trends in Approximation Theory, pp. 203–212. Vanderbilt University Press, Nashville (2001)
Kushpel, A., Tozoni, S.A.: Sharp orders of n-widths of Sobolev’s classes on compact globally symmetric spaces of rank 1. In: 54th Seminário Brasileiro de Análise, São José do Rio Preto, SP, pp. 293–303 (2001)
Kushpel, A., Tozoni, S.A.: On the problem of optimal reconstruction. J. Fourier Anal. Appl. 13(4), 459–475 (2007)
Kwapień, S.: Isomorphic characterizations of inner product spaces by orthogonal series with vector valued coefficients. Stud. Math. 44, 583–595 (1972)
Morimoto, M.: Analytic Functionals on the Sphere. Translations of Mathematical Monographs, vol. 178. Am. Math. Soc., Providence (1998)
Pajor, A., Tomczak-Jaegermann, N.: Subspaces of small codimension of finite-dimensional Banach spaces. Proc. Am. Math. Soc. 97, 637–642 (1986)
Pietsch, A.: Operator Ideals, North-Holland, Amsterdam (1980)
Pinkus, A.: n-Widths in Approximation Theory. Springer, Berlin, (1985)
Pisier, G.: The Volume of Convex Bodies and Banach Space Geometry. Cambridge University Press, London (1989)
Schwartz, L.: Théorie des distributions, Vols. I, II. Hermann, Paris (1950/51)
Sobolev, S.L.: Introduction to the Theory of Cubature Formulas. Nauka, Moscow (1974)
Szegö, G.: Orthogonal Polynomials. Am. Math. Soc., New York (1939)
Triebel, H.: Interpolation Theory, Function Spaces, Differential Operators. VEB/North-Holland, Berlin/Amsterdam (1978)
Triebel, H.: Theory of Function Spaces. Geest & Portig/Birkhäuser, Leipzig/Basel (1983)
Triebel, H.: Theory of Function Spaces II. Birkhäuser, Basel (1992)
Wang, H.C.: Two-point homogeneous spaces. Ann. Math. 55, 177–191 (1952)
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A. Kushpel was supported in part by FAPESP/Brazil, Grant 03/10393-8 and 07/56162-8.
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Communicated by Allan Pinkus.
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Kushpel, A., Tozoni, S.A. Entropy and Widths of Multiplier Operators on Two-Point Homogeneous Spaces. Constr Approx 35, 137–180 (2012). https://doi.org/10.1007/s00365-011-9146-7
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DOI: https://doi.org/10.1007/s00365-011-9146-7