Advances in Pure Mathematics, 2011, 1, 3032 doi:10.4236/apm.2011.12008 Published Online March 2011 (http://www.SciRP.org/journal/apm) Copyright © 2011 SciRes. APM Relative Widths of Some Sets of m p l* Weiwei Xiao, Weijun Luan College of Sciences, North China University of Technology, Beijing, China Email: wwsunny@163.com Received January 14, 2011; revised January 30, 2011; accepted March 10, 2011 Abstract In this paper, the relative widths of some sets in m p l are studied. Relative widths is the further development of Kolmogorov widths and it is a new problem in approximation theory which aroused some mathematics workers great interest recently. We present some basic propositions of relative widths and investigate relative widths of some sets (ball or ellipsoid) of m p l. Keywords: Kolmogorov Widths, Relative Widths 1. Introduction In 1984, V. N. Konovalov in [1] first proposed the definition of relative widths which is in the sense of Kolmogorov. Let W and V be centrally symmetric sets in a Banach space . The Kolmogorov ndimen sional widths of W relative to V in (shortly, relative widths) is ,, :=, sup inf inf nX nn fW LgVL KWVXf g where the infimum is taken over all ndimensional subspaces n L of , nN. When =VX the rela tive widths coincid es with the ndimensional Kolmogo rov widths (shortly, nK widths) of W in , which we denote by , n dWX. Of course, ,, , nn WV XdW X for any set V, and if 12 VV, then 12 ,,,, . nn WV XKWVX Y. N. Subbotin and S. A. Telyakovskii in [79], V. M. Tikhomirov in [11], V. F. Babenko in [24], V. N. Kono valov in [1,5,6], V. T. Shevaldin in [10] etc. gained many results in this field. And some Chinese mathematics workers such as Yongping Liu, Lianhong Yang in [1517] and Weiwei Xiao in [1214] also did some work on rela tive widths. Let m l, 1p, denote space of vect o rs 1 =,, m xx with norm 1 1 =, 1< p pp m pxx p x 1 =max,,, =. m xxp x Let :=: 1 m pp p Bl xx be the unit ball in m l. Let 1 =,, m diag DDD be an mm real diagonal matrix. Without loss of generality we assume that 12 >0 m DD D. Let be a positive real num ber, set =:, , m pp RMDx xx obviously it is ellipsoid in m l. When =1 , we denote it by . Theorem A: [19] For 1p , 1<mn, 1 ,= , m npp n dlD Similar to the proof in [18] we can get the following proposition. Proposition 1. 1) If W is a finite set of elements, then for the linear spanning subspace lin W one has ,,= ,,=0 nn KW lin WXKlin Wlin WX for nm. 2) If 1 WW, then 1,,,, . nn WVXK WVX 3) For any scalar , and any W and V, one has ,,= ,,. nn WVX KWVX 4) 012 ,,,,,, .KWVXK WVXKWVX 5) Let 0 =m WK , where 0 is a bounded set and *Supported partly by National Natural Science Foundation of China (No.10471010) and partly by the project “Representation Theory and Related Topics” of the “985 Program” of Beijing Normal University and partly by Beijing Natural Science Foundation (1062004).
W. W. XIAO ET AL. Copyright © 2011 SciRes. APM 31 m is a subspace of dimension m. If <nm, then ,, = n KWVX . 6) For the convex hull co W, if for each subspace n of dimension n, n co WX is a locally sequen tially compact and closed subset, then ,,= ,,. nn WcoW XKcoWcoW X 7) If Y is a subspace of and WYX , VY, then ,,,, . nn WV XKWVY Theorem 1 For >mnN, 1p , 11 >n DD , the smallest number which makes the equalities 1 ,,= ,=, mm np pp nppn KMldlD (1) hold is 1 01 := 1n D MD , and 10 10 1,0<<, ,,= ,. m np ppn DMM KMl DMM Theorem 2 For all mN such that >1m, 111 1 ,, =. 2 m m KBBl 2. Proof of Theorems Proof of Theorem 1: For 01 =,0,,0Dx, we have 0 1 ,, sup inf inf sup inf inf =1. m np pp nm n x LlyM L p pp p yM xp p p yM p KMl DM xy xy xy That is 1 ,,1, 0<1. m np pp KMlDM M (2) In order to make the equalities (1) hold, we have that 11 1, n DDM that is 1 1 1n D MD . (3) For 1 1 0<1n D MD , we will prove that 1 ,, 1. m np pp Ml MD (4) For each = xDz, 1 pz, set 1 =,,,0,,0 . n np yMx MxLM When =p , the inequality (4) is trivial, so we only need to prove the case of 1<p . 11 111 1 11 1111 11 1 =1 1 11 11. pp ppp p nn m p pp pp nnn nmm pp ppn pp pp ppp nn m ppp pp p Mxx xx DzDzDzD z MDzz MDMDDzz MDz MD xy In fact, when 1 1 0<1n D MD , we have 11 11 11 n MDD . So we get that inequality (4). By inequalities (2) and (4), we have that 11 1 0<1, ,,=1. m nnp pp D KMl MD D (5) From (3) and (5) we get that the smallest number which makes the equalities (1) hold is 1 01 =1 n D MD . For 0 M, 01 ,,, ,=. mm n pppn pppn KMlKMlD (6) By Theorem A, for all >0M, 1 ,, ,=. mm nppp nppn KMldlD (7) From (6) and (7) we get 10 ,,=, . m np ppn Ml DMM The proof of Theorem 1 is complete. Proof of Theorem 2: From [6] we know that 111 1 ,, . 2 m m KBBl (8) We want to prove that 111 1 ,, . 2 m m KBBl (9)
W. W. XIAO ET AL. Copyright © 2011 SciRes. APM 32 By proposition (6) we know that 1111 1 ,,=,, , mm mm BBlK WBl (10) where =0 ,,0,1,0,,0:=1,,, represent the th coordinate i Wimii . Set 112 :=:= 0. mm m LRxxx x For =0,,0,1,0, ,0 i aW, set 11 1 =0, ,0, 12,12,0, ,0, m ii bLB =1, ,id, when =id, 1i represent the 1st coor dinate, we get =12ab . So we proved 11 ,, 12, m m KWBl which means that inequality (9) is valid. The proof of Theorem 2 is complete. 3. References [1] V. N. Konovalov, “Estimates of Diameters of Kolmo gorov Type for Classes of Differentiable Periodic Func tions,” Matematicheskie Zametki, Vol. 35, No. 3, 1984, pp. 369380. [2] V. F. Babenko, “On the Relative Widths of Classes of Functions with Bounded Mixed Derivative,” East Journal on Approximations, Vol. 2, No. 3, 1996, pp. 319330. [3] V. F. Babenko, “Approximation in the Mean under Re striction on the Derivatives of the Approximating Func tions, Questions in Analysis and Approximations,” Insti tute of the Mathematics of the Ukrainean Academy of Science, Kiev, 1989, pp. 918. (in Russian) [4] V. F. Babenko, “On Best Uniform Approximations by Splines in the Presence of Restrictions on Their Deriva tives,” Mathematical Notes, Vol. 50, No. 6, 1991, pp. 12271232. Translated from Matematicheskie Zametki, Vol. 50, No. 6, 1991, pp. 2430. [5] V. N. Konovalov, “Approximation of Sobolev Classes by Their Sections of Fi nite Dimensional,” Ukrainian Mathe matical Journal, Vol. 54, No. 5, 2002, pp. 795805. doi:10.1023/A:1021635530578 [6] V. N. Konovalov, “Approximation of Sobolev Classes by Their FiniteDimensional Sections,” Matematicheskie Za metki, Vol. 72, No. 3, 2002, pp. 370382. [7] Y. N. Subbotin and S. A. Telyakovskii, “Exact Values of Relative Widths of Classes of Differentiable Functions,” Mathematical Notes, Vol. 65, No. 6, 1999, pp. 731738. Translated from Matematicheskie Zametki, Vol. 65, No. 6, 1999, pp. 871879. [8] Y. N. Subbotin and S. A. Telyakovskii, “Relative Widths of Classes of Differentiable Functions in the 2 Met ric,” Russian Mathematical Surveys, Vol. 56, No. 4, 2001, pp. 159160. [9] Y. N. Subbotin and S. A. Telyakovskii, “On Relative Widths of Classes of Differentiable Functions,” Proceed ings of the Steklov Institute of Mathematics, 2005, pp. 243254. [10] V. T. Shevaldin, “Approximation by Local Trigonometric Splines,” Mathematical Notes, Vol. 77, No. 3, 2005, pp. 326334. [11] V. M. Tikhomirov, “Some Remarks on Relative Diame ters,” Banach Center Publications, Vol. 22, 1989, pp. 471474. [12] W. W. Xiao, “Relative Widths of Classes of Functions Defined by a SelfConjugate Linear Differential Operator in q T (in Chinese),” Chinese annals of Mathematics, Vol. 29A, No. 5, 2008, pp. 679688. [13] W. W. Xiao, “Relative InfiniteDimensional Width of Sobolev Classes,” Journal of Mathematical Analysis and Applications, Vol. 369, No. 2, 2010, pp. 575582. doi:10.1016/j.jmaa.2010.03.050 [14] Y. P. Liu and W. W. Xiao, “Relative Average Widths of Sobolev Spaces in 2Rd L,” Analysis Mathematica, Vol. 34, No. 1, 2008, pp. 7182. doi:10.1007/s1047600801078 [15] Y. P. Liu and L. H. Yang, “Relative Width of Smooth Classes of Multivariate Periodic Functions with Restric tions on Interated Laplace Derivatives in The 2 Met Ric,” Acta Mathematica Scientia, Vol. 26B, No. 4, 2006, pp. 720728. doi:10.1016/S02529602(06)600982 [16] Y. P. Liu and L. H. Yang, “Relative Widths of Smooth Factions Determined by Fractional Order Derivatives,” Journal of complexity, Vol. 24, No. 2, 2008, pp. 259282. [17] L. H. Yang and Y. P. Liu, “Relative Widths of Smooth Functions Determined by Linear Differential Operator,” Journal of Mathematical Analysis and Applications, Vol. 351, No. 2, 2009, pp. 734746. doi:10.1016/j.jmaa.2008.11.009 [18] G. G. Lorentz, M. V. Golitschek, and Y. Makovoz, “Con structive Approximation,” Springer, Berlin, 1996. [19] A. Pinkus, “NWidths in Approximation Theory,” Springer, Berlin, 1985.
