We consider the classes of periodic functions with formal self-adjoint linear differential operators Wp(ℒr), 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(ℒr) in the space Lq for 1<p≤q<∞.

1. Introduction and Main Result

Let ℂ, ℝ, ℤ, ℕ, and ℕ+ 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, 1≤q≤∞. 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∈ℕ. The corresponding Sobolev class is the set Wpr∶={x(⋅):x∈W̃pr,‖x(r)(⋅)‖p≤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 formula bn(C,X)∶=supLsup{λ≥0:L∩λBX⊂C},
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 bn(C;X), where C=Wpr and X=Lq, 1≤p, q≤∞. He also obtained the first results [1] for p=q=∞ and n=2k-1. Pinkus [2] found b2n-1(Wpr;Lq), where p=q=1. Later, Magaril-Il'yaev [3] 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. [4] who found the exact values of b2n-1(Wpr;Lq) for all 1<p≤q<∞.

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 ℒr(D). The linear differential operator ℒr(D) will be called formal self-adjoint if pr(-t)=(-1)rpr(t) for each t∈ℂ.

We define the function classes Wp(ℒr) as follows: Wp(Lr)={x(⋅):xr-1∈AC2π,‖Lr(D)x(⋅)‖p≤1},
where 1≤p≤∞.

In this paper, we consider some classes of periodic functions with formal self-adjoint linear differential operators Wp(ℒr), which include the classical Sobolev class as its special case. Using the iterative method of Buslaev and Tikhomirov [5], with the help of the spectrum of linear differential equations, we determine the exact values of Bernstein width of the classes Wp(ℒr) in the space Lq for 1<p≤q<∞. The results of Buslaev et al. [4] 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 ℒr(D) be an arbitrary formal self-adjoint linear differential operator given by (1.3), and n,r∈ℕ, 1<p≤q<∞. 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 ∥ℒr(D)x(·)∥p=1.

2. Proof of the Theorem

First we introduce some notations and formulate auxiliary statements.

Let ℒr(D) be an arbitrary linear differential operator (1.3). Denote the 2π-periodic kernel of ℒr(D) by KerLr(D)={x(⋅)∈Cr(T):Lr(D)x(t)≡0}.
Let μ(0≤μ≤r) be the dimension of Kerℒr(D) and {φi,…,φμ} an arbitrary basis in Kerℒr(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 [2]. Following, (x(·),λ) is called the spectral pair of (1.7) if the function x(·) is normalized by the condition ∥ℒr(D)x(·)∥p=1. The set of all spectral pairs is denoted by SP(p,q,ℒr). Define the spectral classes SP2k(p,q,ℒr) 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)|qdt⟶sup,∫0π/2n|Lr(D)X(t)|pdt≤1,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 t∈𝕋x2n(t)∶={x̂2n(t),0≤t≤π2n,x̂2n(πn-t),π2n<t≤πn.
Let us extend periodically the function x2n(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 x2n(t) belonging to SP2n(p,q,ℒr). Furthermore, by [6], which any other function from SP2n(p,q,ℒr) differs from x2n(·) only in the sign and in a shift of its argument, and there exists a number N∈ℕ+ such that for every n≥N, all zeros of x2n(·) are simple, equidistant with a step equal to π/n, and Sc(x2n)=Sc(ℒr(D)x2n)=2n. We denote the set of zeros (= sign variations) of ℒr(D)x2n on the period by Q2n=(τ1,…,τ2n). Let
Gr(t)=12π∑k∉Λeiktpr(ik),
where Λ={k∈ℤ: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 [7], if n≥N, then dim S(Q2n,Gr)=2n.

We assume (shifting x(·) if necessary) that ℒr(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)+1π∫TGr(t-τ)(∑i=12nbiyi(τ))dτ,
where a1,…,aμ,b1,…,b2n∈ℝ, ∑i=12nbi=0, yi(·)=χi(·)ℒr(D)x2n(·-(i-1)π/n), and χi(·) is the characteristic function of the interval Δi∶=[-π+(i-1)π/n,-π+iπ/n], 1≤i≤2n. Obviously, dim L2n=2n and L2n⊂Wp(ℒr).

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

Let P∈L(X,Y). Then the Bernstein n-width is defined by
bn(P(X),Y)=supXn+1infPx≠0Px∈Xn+1‖Px‖Y‖x‖X,
where Xn+1 is any subspace of span {Px:x∈X} 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(⋅)‖qq‖Lr(D)x(⋅)‖pp⟶inf,x(⋅)∈L2n
and denote the value of this problem by αq. Let us show that α≥λn; this will imply the desired lower bound for b2n-1. Let x(·)∈L2n, then‖Lr(D)x(⋅)‖pp=∑i=12n∫Δi|∑i=12nbiyi(t)|pdt=∑i=12n∫Δi|bi|p|Lr(D)xn(t)|pdt=12n∑i=12n|bi|p
and by setting zi(⋅)∶=1π∫TGr(⋅-τ)yi(τ)dτ,i=1,2,…,2n,

we reduce problem (2.9) to the form ‖∑j=1μajφj(⋅)+∑i=12nbizi(⋅)‖qq1/2n∑i=12n|bi|p⟶inf,a1,…,aμ,b1,…,b2n∈R.

This is a smooth finite-dimensional problem. It has a solution (a¯1,…a¯μ,b¯1,…,b¯2n) and, (b¯1,…,b¯2n)≠0⃗. According to the Lagrange multiplier rule, there exists a η∈ℝ 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 relations∫Tφj(t)(Qqx¯)(t)dt=0,j=1,…,μ,∫Tzi(t)(Qqx¯)(t)dt=12n‖x(⋅)¯‖qq‖Lr(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 d≠0, 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, 1≤i0≤2n.

Let x̃2n(t)=∑j=1μaj⋆φj(t)+∑i=12n(-1)i+1zi(t)
and x̃2n satisfies (1.7). Let a⋆=(a1⋆,…,a2n⋆) and let b⋆=(1,-1,…,1,-1)∈ℝ2n. It follows from the definitions of x̃2n(·) and x¯(·) that Lr(D)x̃2n(t)-Lr(D)x¯(t)=∑i≠i0i=12n((-1)i+1-b¯i)χi(t)Lr(D)x2n(t-(i-1)πn),
and hence Sc(ℒr(D)x̃2n(·),ℒr(D)x¯(·)) has at most 2n-2 sign changes. Then, by Rolle's theorem, Sc(ℒr(D)x̃2n(·)-ℒr(D)x¯(·))≤2n-2. For any a,b∈ℝ, 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, ℒr(D)y(t)=(-1)rλ-q(Qqx)(t) and φj(t)∈Kerℒr(D). Then, by [8, page 94], we have∫Tφj(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, 1≤i≤2n, we get∫Δizi(t)(Qqx̃2n)(t)dt=(-1)i+1∫Δi|x̃2n(t)|qdt=(-1)i+1λ2nq2n,
due to ∫𝕋zi(t)(Qqx̃2n)(t)dt=∫Δizi(t)(Qqx̃2n)(t)dt. Therefore, we have∫Tzi(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)(1π∫TGr(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 ℒr(D)x2n(t-(i-1)π/n) in the integral in the left-hand side of this equality. Since ∥(ℒr(D)x¯)(·)∥p≤1 and hence ∥(ℒr(D)x¯)(·)∥pq-p≤1 for p≤q, if we assume that λ2n>α, then we arrive at the relationssign∫ΔiLr(D)x2n(t-(i-1)πn)f(⋅)dt=(-1)r+i+1,i=1,…,2n.

Suppose for definiteness that ℒr(D)x2n(t-(i-1)π/n)>0 interior to Δi, i=1,…,2n. Then it follows from (2.22) that there are points ti∈Δi 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(ℒr(D)f(·))≥2n. Further, ℒr(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 λ2n≤α. Thus b2n-1(Wp(ℒr);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(ℒr);Lq)>λ2n, (1<p≤q<∞). Then, by definition, there exists a linearly independent system of 2n functions L2n∶=span{f1,…,f2n}⊂Lq and number γ>λ2n such that L2n∩γS(Lq)⊆ℒr(D), or equivalently,minx(⋅)∈L2n‖x(⋅)‖q‖Lr(D)x(⋅)‖p≥γ>λ2n.
Let us assign a vector c∈ℝ2n to each function x(·)∈L2n by the following rule: x(⋅)⟶c=(c1,…,c2n)∈R2n,wherex(⋅)=∑j=12ncjfj(⋅).
Then inequality (2.23) acquires the form minc∈R2n∖{0}‖∑j=12ncjfj(⋅)‖q‖∑j=12ncjLr(D)fj(⋅)‖p≥γ>λ2n.
Let c0=0. Consider the sphere S2n-1 in the space ℝ2n with radius 2π, that is, S2n-1∶={c:c=(c1,…,c2n)∈R2n,‖c‖=∑j=12n|cj|=2π}.
To every vector c∈ℝ2n we assign function u(t,c) defined by u(t,c)={(2π)-1/psignck,fort∈(tk-1,tk),k=1,…,2n,0,fort=tk,k=1,…,2n-1,
where t0=0, tk=∑i=1k|ci|, k=1,…,2n, and the extended 2π-periodically onto ℝ.

An analog of the Buslaev iteration process [5] is constructed in the following way: the function x(t,c) is found as a periodic solution of the linear differential equation ℒr(D)x0=u; then the periodic functions {xk(t,c)}k∈ℕ+ are successively determined from the differential equationsLr(D)xk(t)=(Qp′yk)(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)xk‖p=1,(Qqxk)(t)⊥KerLr(D),(Qp′yk)(t)⊥KerLr(D).

By analogy with the reasoning in [5], we can prove the following assertions.

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

The sequence {xk(·,c)}k∈ℕ 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∈ℕ there exists a ĉ∈S2n-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)=2N≥2n.

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 [10], which states that there exists a ĉ∈S2n-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 [8] the linear differential equation ℒr(D)f=g has a 2π-periodic solution if and only if ∫𝕋g(t)v(t)dt=0, where v(·)∈Kerℒr(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)=λ2N≤λ2n.
Therefore, by virtue of items (i), (ii), and (2.30), we obtain minc∈R2n∖{0}‖∑j=12ncjfj(⋅)‖q‖∑j=12ncjLr(D)fj(⋅)‖p≤‖∑j=12nĉjfj(⋅)‖q‖∑j=12nĉjLr(D)fj(⋅)‖p≤‖xk(⋅,ĉ)‖q‖Lr(D)xk(⋅,ĉ)‖p≤λ(ĉ)≤λ2n,
which contradicts (2.25). Hence b2n-1(Wp(ℒr);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).

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