JAMJournal of Applied Mathematics1687-00421110-757XHindawi Publishing Corporation49505410.1155/2012/495054495054Research ArticleBernstein Widths of Some Classes of Functions Defined by a Self-Adjoint OperatorFengGuo1DiethelmKai1School of Mathematics and Information EngineeringTaizhou UniversityTaizhou 317000Chinatzc.edu.cn201215102011201209052011020820112012Copyright © 2012 Guo Feng.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.

We consider the classes of periodic functions with formal self-adjoint linear differential operators Wp(r), which include the classical Sobolev class as its special case. Using the iterative method of Buslaev, with the help of the spectrum of linear differential equations, we determine the exact values of Bernstein width of the classes Wp(r) in the space Lq for 1<pq<.

1. Introduction and Main Result

Let , , , , and + 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 Lq=Lq[0,2π] be the classical Lebesgue integral space of 2π-periodic real-valued functions with the usual norm ·q, 1q. Denote by W̃pr 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)(·)Lp, r. The corresponding Sobolev class is the set Wpr={x():xW̃pr,x(r)()p1}.

Tikhomirov  introduced the notion of Bernstein width of a centrally symmetric set C in a normed space X. It is defined by the formula bn(C,X)=supLsup{λ0:LλBXC}, where BX is the unit ball of X and the outer supremum is taken over all subspaces LX such that dim Ln+1, n.

In particular, Tikhomirov posed the problem of finding the exact value of bn(C;X), where C=Wpr and X=Lq, 1p, q. He also obtained the first results  for p=q= and n=2k-1. Pinkus  found b2n-1(Wpr;Lq), where p=q=1. Later, Magaril-Il'yaev  obtained the exact value of b2n-1(Wpr;Lq) for 1<p=q<. The latest contribution to these fields is due to Buslaev et al.  who found the exact values of b2n-1(Wpr;Lq) for all 1<pq<.

Let Lr(D)=Dr+ar-1Dr-1++a1D+a0,D=ddt, be an arbitrary linear differential operator of order r with constant real coefficients a0,a1,,ar-1. Denote by pr the characteristic polynomial of r(D). The linear differential operator r(D) will be called formal self-adjoint if pr(-t)=(-1)rpr(t) for each t.

We define the function classes Wp(r) as follows: Wp(Lr)={x():xr-1AC2π,Lr(D)x()p1}, where 1p.

In this paper, we consider some classes of periodic functions with formal self-adjoint linear differential operators Wp(r), which include the classical Sobolev class as its special case. Using the iterative method of Buslaev and Tikhomirov , with the help of the spectrum of linear differential equations, we determine the exact values of Bernstein width of the classes Wp(r) in the space Lq for 1<pq<. The results of Buslaev et al.  are extended to the classes (1.4) defined by differential operators (1.3).

We define Qq to be the nonlinear transformation (Qqf)(t)=|f(t)|q-1signf(t).

The main result of this paper is the following.

Theorem 1.1.

Let r(D) be an arbitrary formal self-adjoint linear differential operator given by (1.3), and n,r, 1<pq<. Then b2n-1(Wp(Lr);Lq)=λ2n=λ2n(p,q,Lr), where λ2n is that eigenvalue λ of the boundary value problem Lr(D)y(t)=(-1)rλ-q(Qqx)(t),y(t)=(QpLr(D)x)(t),x(j)(0)=x(j)(2π),y(j)(0)=y(j)(2π),j=0,1,,n-1, for which the corresponding eigenfunction x(·)=x2n(·) has only 2n simple zeros on 𝕋 and is normalized by the condition r(D)x(·)p=1.

2. Proof of the Theorem

First we introduce some notations and formulate auxiliary statements.

Let r(D) be an arbitrary linear differential operator (1.3). Denote the 2π-periodic kernel of r(D) by KerLr(D)={x()Cr(T):Lr(D)x(t)0}. Let μ(0μr) be the dimension of Kerr(D) and {φi,,φμ} an arbitrary basis in Kerr(D).

Zc(f) denotes the number of zeros of f on a period, counting multiplicity, and Sc(f) is the cyclic sign change count for a piecewise continuous, 2π-Periodic function f . Following, (x(·),λ) is called the spectral pair of (1.7) if the function x(·) is normalized by the condition r(D)x(·)p=1. The set of all spectral pairs is denoted by SP(p,q,r). Define the spectral classes SP2k(p,q,r) as SP2k(p,q,Lr)={(x(),λ)SP(p,q,Lr):Sc(x())=2k}.

Let x̂2n(·) be the solution of the extremal problem as follows: 0π/2n|X(t)|qdtsup,0π/2n|Lr(D)X(t)|pdt1,x(k)(((π/2n)+(-1)k+1(π/2n))2)=0,k=0,1,,n-1 and the function x2n(·) is such that x2n(t)=-x2n(t-π/n) for all t𝕋x2n(t)={x̂2n(t),0tπ2n,x̂2n(πn-t),π2n<tπn. Let us extend periodically the function x2n(t) onto 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 x2n(t) belonging to SP2n(p,q,r). Furthermore, by , which any other function from SP2n(p,q,r) differs from x2n(·) only in the sign and in a shift of its argument, and there exists a number N+ such that for every nN, all zeros of x2n(·) are simple, equidistant with a step equal to π/n, and Sc(x2n)=Sc(r(D)x2n)=2n. We denote the set of zeros (= sign variations) of r(D)x2n on the period by Q2n=(τ1,,τ2n). Let Gr(t)=12πkΛeiktpr(ik), where Λ={k:pr(ik)=0} and i is the imaginary unit.

The 2π-periodic G-splines are defined as elements of the linear space S(Q2n,Gr)=span{φ1(t),,φμ(t),Gr(t-τ1),,Gr(t-τ2n)}. As was proved in , if nN, then dim S(Q2n,Gr)=2n.

We assume (shifting x(·) if necessary) that r(D)x̂2n(·) is positive on (-π,π+π/n). Let L2n:=L2n(r,p,q) denote the space of functions of the form x(t)=j=1μajφj(t)+1πTGr(t-τ)(i=12nbiyi(τ))dτ, where a1,,aμ,  b1,,b2n, i=12nbi=0, yi(·)=χi(·)r(D)x2n(·-(i-1)π/n), and χi(·) is the characteristic function of the interval Δi=[-π+(i-1)π/n,-π+iπ/n], 1i2n. Obviously, dim L2n=2n and L2nWp(r).

Let us now consider exact estimate of Bernstein n-width. This was introduced in . We reformulate the definition for a linear operator P mapping X to Y.

Definition 2.1 (see [<xref ref-type="bibr" rid="B9">2</xref>, page 149]).

Let PL(X,Y). Then the Bernstein n-width is defined by bn(P(X),Y)=supXn+1infPx0PxXn+1PxYxX, where Xn+1 is any subspace of span {Px:xX} of dimension n+1.

2.1. Lower Estimate of Bernstein <bold><inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M144"><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:math></inline-formula></bold>-Width

Consider the extremal problem x()qqLr(D)x()ppinf,x()L2n and denote the value of this problem by αq. Let us show that αλn; this will imply the desired lower bound for b2n-1. Let x(·)L2n, thenLr(D)x()pp=i=12nΔi|i=12nbiyi(t)|pdt=i=12nΔi|bi|p|Lr(D)xn(t)|pdt=12ni=12n|bi|p and by setting zi()=1πTGr(-τ)yi(τ)dτ,i=1,2,,2n,

we reduce problem (2.9) to the form j=1μajφj()+i=12nbizi()qq1/2ni=12n|bi|pinf,a1,,aμ,b1,,b2nR.

This is a smooth finite-dimensional problem. It has a solution (a¯1,a¯μ,b¯1,,b¯2n) and, (b¯1,,b¯2n)0. According to the Lagrange multiplier rule, there exists a η such that the derivatives of the function (a1,,aμ,b1,,b2n)g(a1,,aμ,b1,,b2n)+η(b1+b2++b2n) (where g(·) is the function being minimized in (2.12)) with respect to a1,,aμ,b1,,b2n at the point (a¯1,,a¯μ,b¯1,,b¯2n) are equal to zero. This leads to the relationsTφj(t)(Qqx¯)(t)dt=0,j=1,,μ,Tzi(t)(Qqx¯)(t)dt=12nx()¯qqLr(D)x¯()ppQpb¯i,i=1,,2n, where x¯(·)=j=1μa¯jφj(t)+i=12nb¯izi(·).

We remark that g(a1,,aμ,b1,,b2n)=g(da1,,daμ,db1,,db2n) for any d0, and hence the vector (da¯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 and b¯i0=(-1)i0+1 for some i0, 1i02n.

Let x̃2n(t)=j=1μajφj(t)+i=12n(-1)i+1zi(t) and x̃2n satisfies (1.7). Let a=(a1,,a2n) and let b=(1,-1,,1,-1)2n. It follows from the definitions of x̃2n(·) and x¯(·) that Lr(D)x̃2n(t)-Lr(D)x¯(t)=ii0i=12n((-1)i+1-b¯i)χi(t)Lr(D)x2n(t-(i-1)πn), and hence Sc(r(D)x̃2n(·),  r(D)x¯(·)) has at most 2n-2 sign changes. Then, by Rolle's theorem, Sc(r(D)x̃2n(·)-r(D)x¯(·))2n-2. For any a,b, sign(a+b)=sign(Qpa+Qpb); therefore Sc((Qqx̃2n)()-(Qqx¯)())2n-2.

In addition, since x̃2n is 2π-periodic solution of the linear differential equation, r(D)y(t)=(-1)rλ-q(Qqx)(t) and φj(t)Kerr(D). Then, by [8, page 94], we haveTφj(t)(Qqx̃)(t)dt=0,j=1,,μ.

If we now multiply both sides of (2.15) by (Qqx̃2n)(t) and integrate over the interval Δi, 1i2n, we getΔizi(t)(Qqx̃2n)(t)dt=(-1)i+1Δi|x̃2n(t)|qdt=(-1)i+1λ2nq2n, due to 𝕋zi(t)(Qqx̃2n)(t)dt=Δizi(t)(Qqx̃2n)(t)dt. Therefore, we haveTzi(t)(Qqx̃2n)(t)dt=(-1)i+1λ2nq2n,i=1,,2n. Changing the order of integration and using (2.14) and (2.20), we get that ΔiLr(D)x2n(t-(i-1)πn)(1πTGr(t-τ)((Qqx̃2n)(τ)-(Qqx¯)(τ))dτ)dt=Tzi(t)((Qqx̃2n)(t)-(Qqx¯)(t))dt=(-1)r2n((-1)i+1λ2nq-x()¯qq(Lr(D)x¯)()ppQpb¯i)=(-1)r2n((-1)i+1λ2nq-αq(Lr(D)x¯)()q-pQpbi¯),i=1,,2n. Denote by f(·) the factor multiply r(D)x2n(t-(i-1)π/n) in the integral in the left-hand side of this equality. Since (r(D)x¯)(·)p1 and hence (r(D)x¯)(·)pq-p1 for pq, if we assume that λ2n>α, then we arrive at the relationssignΔiLr(D)x2n(t-(i-1)πn)f()dt=(-1)r+i+1,i=1,,2n.

Suppose for definiteness that r(D)x2n(t-(i-1)π/n)>0 interior to Δi, i=1,,2n. Then it follows from (2.22) that there are points tiΔi such that signf(ti)=(-1)i+1, i=1,,2n, that is, Sc(f(·))2n-1. But f(·) is periodic, and hence Sc(f(·))2n; therefore, Sc(r(D)f(·))2n. Further, r(D)f(·)=(Qqx̃2n)(t)-(Qqx¯)(t), that is, Sc((Qqx̃2n)(t)-(Qqx¯)(t))2n.

We have arrived at a contradiction to (2.17), and hence λ2nα. Thus b2n-1(Wp(r);Lq)λ2n.

2.2. Upper Estimate of Bernstein <bold><inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M217"><mml:mrow><mml:mi>n</mml:mi></mml:mrow></mml:math></inline-formula></bold>-Width

Assume the contrary: b2n-1(Wp(r);Lq)>λ2n, (1<pq<). Then, by definition, there exists a linearly independent system of 2n functions L2n=span{f1,,f2n}Lq and number γ>λ2n such that L2nγS(Lq)r(D), or equivalently,minx()L2nx()qLr(D)x()pγ>λ2n. Let us assign a vector c2n to each function x(·)L2n by the following rule: x()c=(c1,,c2n)R2n,where  x()=j=12ncjfj(). Then inequality (2.23) acquires the form mincR2n{0}j=12ncjfj()qj=12ncjLr(D)fj()pγ>λ2n. Let c0=0. Consider the sphere S2n-1 in the space 2n with radius 2π, that is, S2n-1={c:c=(c1,,c2n)R2n,  c=j=12n|cj|=2π}. To every vector c2n we assign function u(t,c) defined by u(t,c)={(2π)-1/psignck,  for  t(tk-1,tk),  k=1,,2n,0,  for  t=tk,  k=1,,2n-1, where t0=0, tk=i=1k|ci|, k=1,,2n, and the extended 2π-periodically onto .

An analog of the Buslaev iteration process  is constructed in the following way: the function x(t,c) is found as a periodic solution of the linear differential equation r(D)x0=u; then the periodic functions {xk(t,c)}k+ are successively determined from the differential equationsLr(D)xk(t)=(Qpyk)(t),Lr(D)yk(t)=(-1)rμk-1-q(Qqxk-1)(t), where p=p/(p-1), and the constants {μk:k=0,} are uniquely determined by the conditions Lr(D)xkp=1,(Qqxk)(t)KerLr(D),(Qpyk)(t)KerLr(D).

By analogy with the reasoning in , we can prove the following assertions.

The iteration procedure (2.28)-(2.29) is well defined; the sequences {μk}k are monotone nondecreasing and converge to an eigenvalue λ(c)>0 of the problem (1.7).

The sequence {xk(·,c)}k has a subsequence that is convergent to an eigenfunction x(·,c) of the problem (1.7), with λ(c)=x(·,c)p.

For any k there exists a ĉS2n-1 such that xk(·,ĉ) has at least 2n zeros (Zc(xk(·,ĉ))2n) on 𝕋.

In the set of spectral pairs (λ(c),x(·,c)), there exists a pair (λ(ĉ),x(·,ĉ)) such that Sc(x(·,ĉ)=2N2n.

Items (i) and (ii) can be proved in the same way as Lemmas 1 and 2 of [5, Sections 6 and 10]. Item (iii) follows from the Borsuk theorem , which states that there exists a ĉS2n-1 such that Zc(xk(·,ĉ))2n-1, but since the function xk(·,ĉ) is periodic, we actually have Zc(xk(·,ĉ))2n. Finally, in item (iv), by (ii) and (iii), Zc(x(·,ĉ))2n. In view of x(·,ĉ) zeros are simple; therefore, Sc(x(·,ĉ))2n.

Note that  the linear differential equation r(D)f=g has a 2π-periodic solution if and only if 𝕋g(t)v(t)dt=0, where v(·)Kerr(D) and g is an integrable 2π-periodic function. Using the method similar to [5, 11], it is not difficult to show that spectral pairs of (1.7) are unique and spectral value λn is monotone decreasing for n; it follows thatλ(ĉ)=λ2Nλ2n. Therefore, by virtue of items (i), (ii), and (2.30), we obtain mincR2n{0}j=12ncjfj()qj=12ncjLr(D)fj()pj=12nĉjfj()qj=12nĉjLr(D)fj()pxk(,ĉ)qLr(D)xk(,ĉ)pλ(ĉ)λ2n, which contradicts (2.25). Hence b2n-1(Wp(r);Lq)λ2n. Thus, the upper bound is proved. This completes the proof of the theorem.

Acknowledgments

The author would like to thank Professor Kai Diethelm and the anonymous referees for their valuable comments, remarks, and suggestions which greatly help us to improve the presentation of this paper and make it more readable. Project Supported by the Natural Science Foundation of China (Grant no. 10671019) and Scientific Research fund of Zhejiang Provincial Education Department (Grant no. 20070509).

TikhomirovV. M.Some Questions in Approximation Theory1976Moscow, RussiaIzdat. Moskov. Univ.0487161PinkusA.n-Widths in Approximation Theory1985New York, NY, USASpringer774404Magaril-Il'yaevG. G.Mean dimension, widths, and optimal recovery of Sobolev classes of functions on the lineMathematics of the USSR—Sbornik1993742381403BuslaevA. P.Magaril-Il'yaevG. G.Nguen T'en NamExact values of Bernstein widths for Sobolev classes of periodic functionsMatematicheskie Zametki199558113914310.1007/BF023061871361119BuslaevA. P.TikhomirovV. M.Spectra of nonlinear differential equations and widths of Sobolev classesMathematics of the USSR—Sbornik1992712427446NovikovS. I.Exact values of widths for some classes of periodic functionsThe East Journal on Approximations19984135541613786Nguen Thi Thien HoaOptimal quadrature formulae and methods for recovery on function classds defined by variation diminishing convolutions, Candidate's Dissertation1985Moscow, RussiaMoscow State UniversityJakubovitchV. A.StarzhinskiV. I.Linear Differential Equations with Periodic Coeflicients and Its Applications1972Moscow, RussiaNaukaPinkusA.n-widths of Sobolev spaces in LpConstructive Approximation198511156210.1007/BF01890021766094ZBL0582.41018BorsukK.Drei Sätze über die n-dimensionale euklidische SphäreFundamenta Mathematicae193320177190ZBL0006.42403BuslaevA. P.TikhomirovV. M.Some problems of nonlinear analysis and approximation theorySoviet Mathematics—Doklady198528311318796951