Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2008, Article ID 894529, 8 pages doi:10.1155/2008/894529 Research Article Exact Values of Bernstein n-Widths for Some Classes of Periodic Functions with Formal Self-Adjoint Linear Differential Operators Feng Guo Department of Mathematics, Taizhou University, Zhejiang, Taizhou 317000, China Correspondence should be addressed to Feng Guo, guofeng0576@163.com Received 10 December 2007; Accepted 18 June 2008 Recommended by Vijay Gupta We consider the classes of periodic functions with formal self-adjoint linear differential operators W p L r , which include the classical Sobolev class as its special case. With the help of the spectral of linear differential equations, we find the exact values of Bernstein n-width of the classes W p L r  in the L p for 1 <p<∞. Copyright q 2008 Feng Guo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1. Introduction and main result Let C, R, Z, N,andN  be the sets of all complex numbers, real numbers, integers, nonnegative integers, and positive integers, respectively. Let T be the unit circle realized as the interval 0, 2π with the points 0 and 2π identified, and as usual, let L q : L q 0, 2π be the classical Lebesgue integral space of 2π-periodic real-valued functions with the usual norm · q , 1 ≤ q ≤ ∞.Denoteby  W r p the Sobolev space of functions x· on T such that the r − 1st derivative x r−1 · is absolutely continuous on T and x r · ∈ L p ,r∈ N. The corresponding Sobolev class is the set W r p :   W r p : x r · p ≤ 1  . 1.1 Tikhomirov 1 introduced the notion of Bernstein width of a centrally symmetric set C in a normed space X. It is defined by the following formula: b n C, X : sup L sup{λ ≥ 0:L ∩ λBX ⊂ C}, 1.2 2 Journal of Inequalities and Applications where BX is the unit ball of X and the outer supremum is taken over all subspaces L ⊂ X such that dim L ≥ n  1,n∈ N. In particular, Tikhomirov posed the problem of finding the exact value of b n C; X,where C  W r p and X  L q , 1 ≤ p, q ≤∞. He also obtained the first results 1 for p  q  ∞ and n  2k − 1. Pinkus 2 found b 2n−1 W r p ; L q ,wherep  q  1. Later, Magaril-Il’yaev 3 obtained the exact value of b 2n−1 W r p ; L p , for 1 <p<∞. The latest contribution to this fields is due to Buslaevetal.4 who found the exact values of b 2n−1 W r p ; L q  for all 1 <p≤ q<∞. Let L r DD r  a r−1 D r−1  ··· a 1 D  a 0 ,D d dt , 1.3 be an arbitrary linear differential operator of order r with constant real coefficients a 0 ,a 1 , ,a r−1 .Denotebyp r the characteristic polynomial of L r D. The linear differential operator L r D will be called formal self-adjoint if p r −t−1 r p r t, for each t ∈ C. We define the function classes W p L r  as follows: W p  L r    x· : x r−1 ∈ AC 2π , L r Dx· p ≤ 1  , 1.4 where 1 ≤ p ≤∞. In this paper, we will determine the exact values of Bernstein n-width of some classes of periodic functions with formal self-adjoint linear differential operators W p L r , which include the classical Sobolev class as its special case. We define Q p to be the nonlinear transformation  Q p f  t :   ft   p−1 signft. 1.5 The maim result of this paper is the following. Theorem 1.1. Assume that 1 <p<∞.LetL r D be an arbitrary formal self-adjoint linear differential operators given by 1.3. Then, there exists a number N ∈ N  such that for every n ≥ N: b 2n−1  W p L r  ; L p λ 2n : λ 2n  p, p,L r  , 1.6 where λ 2n is that eigenvalue λ of the boundary value problem L r Dyt−1 r λ −p  Q p x  t, yt  Q p L r Dx  t, x j 0x j 2π,y j 0y j 2π,j 0, 1, ,n− 1, 1.7 for which the corresponding eigenfunction x·x 2n · has only 2n simple zeros on T and is normalized by the condition L r Dx· p  1. Feng Guo 3 2. Proof of the theorem First we introduce some notations and formulate auxiliary statements. Let L r D be an arbitrary linear differential operator 1.3.Denotethe2π-periodic kernel of L r D by KerL r D  x· ∈ C r T : L r Dxt ≡ 0  . 2.1 Let μ 0 ≤ μ ≤ r be the dimension of KerL r D and {ϕ i , ,ϕ μ } an arbitrary basis in KerL r D. Z c f denotes the number of zeros of f in a period, counting multiplicity, and S c f is the cyclic sign change count for a piecewise continuous, 2π-periodic function f 2. Following, x·,λ is called the spectral pair of 1.7 if the function x· is normalized by the condition L r Dx· p  1. The set of all spectral pairs is denoted by SPp, p, L r . Define the spectral classes SP 2k p, p,L r  as SP 2k p, p,L r   x·,λ  ∈ SP  p, p,L r  : S c  x·   2k  . 2.2 Let x 2n · denotes the solution of the extremal problem as follows:  π/2n 0 |Xt| p dt −→ sup,  π/2n 0 |L r DXt| p dt ≤ 1, x k  π 2n −1 k1 π 2n  /2   0,k 0, 1, ,n− 1, 2.3 and the function x 2n · is such that x 2n t−x 2n t − π/n for all t ∈ T: x 2n t : ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ x 2n t, 0 ≤ t ≤ π 2n , x 2n  π n − t  , π 2n <t≤ π n . 2.4 Let us extend periodically the function x 2n t onto R, and normalize the obtained function as it is required in the definition of spectral pairs. From what has been done above, we get a function x 2n t belongs to SP 2n p, p,L r . Furthermore, by 5, which any other function from SP 2n p, p,L r  differs from x 2n · only in the sign and in a shift of its argument, and there exists a number N ∈ N  such that for every n ≥ N,allzerosofx 2n · are simple, equidistant w ith a step equal to π/n,andS c x 2n S c L r Dx 2n 2n. We denote the set of zeros  sign variations of L r Dx 2n on the period by Q 2n τ 1 , ,τ 2n .Let G r t 1 2π  k / ∈Λ e ikt p r ik , 2.5 where Λ{k ∈ Z : p r ik0} and i is the imaginary unit. 4 Journal of Inequalities and Applications The 2π-periodic G-splines are defined as elements of the linear space S  Q 2n ,G r   span  ϕ 1 t, ,ϕ μ t,G r  t − τ 1  , ,G r  t − τ 2n  . 2.6 As was proved in 6,ifn ≥ N,thendimSQ 2n ,G r 2n. We assume shifting x· if necessary that L r Dx 2n · is positive on −π, π  π/n.Let L 2n : L 2n r, p, p denote the space of functions of the form xt μ  j1 a j ϕ j t 1 π  T G r t − τ  2n  i1 b i y i τ  dτ, 2.7 where a 1 , ,a μ ,b 1 , ,b 2n ∈ R,  2n i1 b i  0,y i ·χ i ·L r Dx 2n ·−i − 1π/n,andχ i · is the characteristic function of the interval Δ i :−π i − 1π/n,−π  iπ/n, 1 ≤ i ≤ 2n. Obviously, dim L 2n  2n and L 2n ⊂ W p L r . Let us now consider exact estimate of Bernstein n-width. This was introduced in 1.We reformulate the definition for a linear operator P mapping X to Y . Definition 2.1 see 2, page 149.LetP ∈ LX, Y . Then the Bernstein n-width is defined by b n PX,Ysup X n1 inf Px∈X n1 Px /  0 Px Y x X , 2.8 where X n1 is any subspace of span {Px : x ∈ X} of dimension ≥ n  1. 2.1. Lower estimate of Bernstein n-width Consider the extremal problem x· p p L r Dx· p p −→ inf,x· ∈ L 2n , 2.9 anddenotethevalueofthisproblembyα p . Let us show that α ≥ λ n , this will imply the desired lower bound for b 2n−1 .Letx· ∈ L 2n ,then L r Dx· p p  2n  i1  Δ i      2n  i1 b i y i t      p dt  2n  i1  Δ i |b i | p |L r Dx n t| p dt  1 2n 2n  i1 |b i | p , 2.10 and by setting z i · : 1 π  T G r ·−τy i τdτ, i  1, 2, ,2n, 2.11 we reduce problem 2.9 to the form   μ j1 a j ϕ j ·  2n i1 b i z i · p p 1/2n  2n i1 |b i | p −→ inf,a 1 , ,a μ ,b 1 , ,b 2n ∈ R. 2.12 Feng Guo 5 This is a smooth finite-dimensional problem. It has a solution  a 1 , a μ , b 1 , ,b 2n ,and, moreover, b 1 , ,b 2n  /  0. According to the Lagrange multiplier rule, there exists a η ∈ R such that the derivatives of the function a 1 , ,a μ ,b 1 , ,b 2n  → ga 1 , ,a μ ,b 1 , ,b 2n  ηb 1  b 2  ···  b 2n where g· is the function being minimized in2.12 with respect to a 1 , ,a μ ,b 1 , ,b 2n at the point a 1 , a μ , b 1 , ,b 2n  are equal to zero. This leads to the relations  T ϕ j t  Q p x  tdt  0,j 1, ,μ, 2.13  T z i t  Q p x  tdt  α p 2n Q p b i ,i 1, ,2n, 2.14 where x·  μ j1 a j ϕ j t  2n i1 b i z i ·. We remark that ga 1 , ,a μ ,b 1 , ,b 2n gda 1 , ,da μ ,db 1 , ,db 2n  for any d /  0, and hence the vector d a 1 , ,da μ ,db 1 , ,db 2n  is also a solution of 2.12. Thus, it can be assumed that |b i |≤1,i 1, ,2n,andb i 0 −1 i 0 1 for some i 0 , 1 ≤ i 0 ≤ 2n. Let x 2n t μ  j1 a  j ϕ j t 2n  i1 −1 i1 z i t, 2.15 and x 2n satisfies 1.7.Leta  a  1 , ,a  2n  and b  1, −1, ,1, −1 ∈ R 2n . It follows from the definitions of x 2n · and x· that L r Dx 2n t −L r Dxt 2n  i1 i /  i 0  −1 i1 − b i  χ i tL r Dx 2n  t − i − 1π n  , 2.16 and hence S c L r Dx 2n ·, L r Dx· has at most 2n−2 sign changes. Then, by Rolle’s theorem, S c L r Dx 2n · −L r Dx· ≤ 2n − 2. For any a, b ∈ R, signa  bsignQ p a  Q p b,therefore S c  Q p x 2n  · −  Q p x·  ≤ 2n − 2. 2.17 In addition, since x 2n is 2π-periodic solution of the linear differential equation L r Dyt−1 r λ −p Q p xt,andϕ j t ∈ KerL r D. Then, by 7, page 94,wehave  T ϕ j t  Q p x  tdt  0,j 1, ,μ. 2.18 If we now multiply both sides of 2.15 by Q p x 2n t, and integrate over the interval Δ i , 1 ≤ i ≤ 2n,weget  Δ i z i t  Q p x 2n  tdt −1 i1  Δ i |x 2n t| p dt −1 i1 λ p 2n 2n . 2.19 Due to  T z i t  Q p x 2n  tdt   Δ i z i t  Q p x 2n  tdt.Therefore,wehave  T z i t  Q p x 2n  tdt −1 i1 λ p 2n 2n ,i 1, ,2n. 2.20 6 Journal of Inequalities and Applications Changing the order of integration and using 2.14 and 2.20,wegetthat  Δ i L r Dx 2n  t − i − 1π n  1 π  T G r t − τ  Q p x 2n  τ −  Q p x  τ  dτ  dt   T z i t  Q p x 2n  t −  Q p xt  dt  1 2n  −1 i1 λ p 2n − α p Q p b i  . 2.21 Denote by f· the factor multiply L r Dx 2n t − i − 1π/n in the integral in the left-hand side of this equality. If we assume that λ 2n >α, then we arrive at the relations sign  Δ i L r Dx 2n  t − i − 1π n  f·dt −1 i1 ,i 1, ,2n. 2.22 Suppose for definiteness that L r Dx 2n t − i − 1π/n > 0interiortoΔ i ,i 1, ,2n. Then it follows from 2.22 that there are points t i ∈ Δ i such that signft i −1 i1 ,i 1, ,2n,thatis,S c f· ≥ 2n − 1. But f· is periodic, and hence S c f· ≥ 2n, therefore, S c L r Df· ≥ 2n. Further, L r Df·  Q p x 2n  t − Q p xt,thatis,S c Q p x 2n t −  Q p xt  ≥ 2n. We have arrived at a contradiction to 2.17, and hence λ 2n ≤ α.Thusb 2n−1 W p L r ; L p  ≥ λ 2n . 2.2. Upper estimate of Bernstein n-width Assume the contrary: b 2n−1 W p L r ; L p  >λ 2n , 1 <p<∞. Then, by definition, there exists a linearly independent system of 2n functions L 2n : span  f 1 , ,f 2n  ⊂ L p and number γ>λ 2n such that L 2n ∩ γSL p  ⊆L r D, or equivalently, min x·∈L 2n x· p L r Dx· p ≥ γ>λ 2n . 2.23 Let us assign a vector c ∈ R 2n to each function x· ∈ L 2n by the following rule: x· −→ c c 1 , ,c 2n  ∈ R 2n , where x· 2n  j1 c j f j ·. 2.24 Then 2.23 acquires the form min c∈R 2n \{0}   2n j1 c j f j · p   2n j1 c j L r Df j · p ≥ γ>λ 2n . 2.25 Let c 0  0. Consider the sphere S 2n−1 in the space R 2n with radius 2π,thatis, S 2n−1 :  c : c c 1 , ,c 2n  ∈ R 2n , c  2n  j1 |c j |  2π  . 2.26 Feng Guo 7 To every vector c ∈ R 2n we assign function ut, c defined by ut, c ⎧ ⎨ ⎩ 2π −1/p sign c j , for t ∈  t k−1 ,t k  ,k 1, ,2n, 0, for t  t k ,k 1, ,2n − 1, 2.27 where t 0  0,t k   k i1 |c i |,k 1, ,2n, and the extended 2π-periodically onto R. An analog of the Buslaev iteration process 8 is constructed in the following way: the function xt, c is found as a periodic solution of the linear differential equation L r Dx 0  u, then the periodic functions {x k t, c} k∈N  are successively determined from the differential equations L r Dx k t  Q p  y k  t, L r Dy k t−1 r μ −p k−1  Q p  x k−1  t, 2.28 where p   p/p − 1, and the constants {μ k : k  0 , ,} are uniquely determined by the conditions L r Dx k  p  1,  Q p x k  t ⊥ KerL r D,  Q p  y k  t ⊥ KerL r D. 2.29 By analogy with the reasoning in 8, we can prove the following assertions: i the iteration procedure 2.28-2.29 is well de fined, the sequences {μ k } k∈N is monotone nondecreasing and converge to an eigenvalue λc > 0 of the problem 1.7, ii the sequence {x k ·,c} k∈N has a subsequence that is convergent to an eigenfunction x·,c of the problem 1.7,withλcx·,c p , iii for any k ∈ N there exists a c ∈ S 2n−1 such that x k ·, c has at least 2n zeros Z c x k ·, c ≥ 2n on T, iv in the set of spectral pairs λc,x·,c, there exists a pair λc,x·, c such that S c x·, c2N ≥ 2n. Items i and ii can be proved in the same way as 8, Sections 6 and 10. Item iii follows from the Borsuk theorem 9, which states that there exists a c ∈ S 2n−1 such that Z c x k ·, c ≥ 2n−1, but since the function x k ·, c is periodic, we actually have Z c x k ·, c ≥ 2n. Finally, item iv,byii and iii, which Z c x·, c ≥ 2n.Inviewofx·, c zeros are simple, therefore, S c x·, c ≥ 2n. Since spectral pairs of 1.7 are unique and the Kolmogorov width d 2n W p L r ; L q  λ 2n p, q, L r  for p ≥ q 5,whenn ≥ N, it follows that λcλ 2N  d 2N  W p  L r  ; L p  ≤ d 2n  W p  L r  ; L p   λ 2n . 2.30 Therefore, by virtue of items i, ii,and2.30,weobtain min c∈R 2n \{0}   2n j1 c j f j · p   2n j1 c j L r Df j · p ≤   2n j1 c j f j · p   2n j1 c j L r Df j · p ≤ x k ·, c p L r Dx k ·, c p ≤ λcλ 2N ≤ λ 2n , 2.31 which contradicts 2.25. Hence b 2n−1 W p L r ; L p  ≤ λ 2n . Thus, the upper bound is proved. This completes the proof of the theorem. 8 Journal of Inequalities and Applications Acknowledgments Project was supported by the Natural Science Foundation of China Grant no. 10671019 and Scientific Research Fund of Zhejiang Provincial Education Department Grant no. 20070509. References 1 V. M. Tikhomirov, Some Questions in Approximation Theory, Izdatel’stvo Moskovskogo Universiteta, Moscow, Russia, 1976. 2 A. Pinkus, n-Widths in Approximation Theory, vol. 7 of Results in Mathematics and Related Areas 3, Springer, Berlin, Germany, 1985. 3 G. G. Magaril-Il’yaev, “Mean dimension, widths and optimal recovery of Sobolev classes of functions on a straight line,” Mathematics of the USSR Sbornik, vol. 74, no. 2, pp. 381–403, 1993. 4 A. P. Buslaev, G. G. Magaril-Il’yaev, and N. T’en Nam, “Exact values of Bernstein widths of Sobolev classes of periodic functions,” Matematicheskie Zametki, vol. 58, no. 1, pp. 139–143, 1995 Russian. 5 S. I. Novikov, “Exact values of widths for some classes of periodic functions,” East Journal on Approximations, vol. 4, no. 1, pp. 35–54, 1998. 6 N. T. T. Hoa, Optimal quadrature formulae and methods for recovery on function Classds defined by variation diminishing convolutions, Candidate’s dissertation, Moscow State University, Moscow, Russia, 1985. 7 V. A. Jakubovitch and V. I. Starzhinski, Linear Differential Equations with Periodic Coeflicients and Its Applications, Nauka, Moscow, Russia, 1972. 8 A. P. Buslaev and V. M. Tikhomirov, “Spectra of nonlinear differential equations and widths of Sobolev classes,” Mathematics of the USSR Sbornik, vol. 71, no. 2, pp. 427–446, 1992. 9 K. Borsuk, “Drei S ¨ atze ¨ uber die n-dimensionale euklidische Sph ¨ are,” Fundamenta Mathematicae, vol. 20, pp. 177–190, 1933. . Corporation Journal of Inequalities and Applications Volume 2008, Article ID 894529, 8 pages doi:10.1155/2008/894529 Research Article Exact Values of Bernstein n-Widths for Some Classes of Periodic Functions with. 1.4 where 1 ≤ p ≤∞. In this paper, we will determine the exact values of Bernstein n-width of some classes of periodic functions with formal self-adjoint linear differential operators W p L r ,. values of Bernstein widths of Sobolev classes of periodic functions, ” Matematicheskie Zametki, vol. 58, no. 1, pp. 139–143, 1995 Russian. 5 S. I. Novikov, Exact values of widths for some classes