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 𝑊𝑟𝑝 the Sobolev space of functions 𝑥() on 𝕋 such that the (𝑟1)st derivative 𝑥(𝑟1)() is absolutely continuous on 𝕋 and 𝑥(𝑟)()𝐿𝑝, 𝑟. The corresponding Sobolev class is the set 𝑊𝑟𝑝𝑊=𝑥()𝑥𝑟𝑝,𝑥(𝑟)()𝑝1.(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 formula 𝑏𝑛(𝐶,𝑋)=sup𝐿sup{𝜆0𝐿𝜆𝐵𝑋𝐶},(1.2) 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=𝑊𝑟𝑝 and 𝑋=𝐿𝑞, 1𝑝, 𝑞. He also obtained the first results [1] for 𝑝=𝑞= and 𝑛=2𝑘1. Pinkus [2] found 𝑏2𝑛1(𝑊𝑟𝑝;𝐿𝑞), where 𝑝=𝑞=1. Later, Magaril-Il'yaev [3] obtained the exact value of 𝑏2𝑛1(𝑊𝑟𝑝;𝐿𝑞) for 1<𝑝=𝑞<. The latest contribution to these fields is due to Buslaev et al. [4] who found the exact values of 𝑏2𝑛1(𝑊𝑟𝑝;𝐿𝑞) for all 1<𝑝𝑞<.

Let 𝑟(𝐷)=𝐷𝑟+𝑎𝑟1𝐷𝑟1++𝑎1𝐷+𝑎0𝑑,𝐷=𝑑𝑡,(1.3) be an arbitrary linear differential operator of order 𝑟 with constant real coefficients 𝑎0,𝑎1,,𝑎𝑟1. 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: 𝑊𝑝𝑟=𝑥()𝑥𝑟1𝐴𝐶2𝜋,𝑟(𝐷)𝑥()𝑝1,(1.4) 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 [5], 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. [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)𝑞1sign𝑓(𝑡).(1.5)

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 𝑏2𝑛1𝑊𝑝𝑟;𝐿𝑞=𝜆2𝑛=𝜆2𝑛𝑝,𝑞,𝑟,(1.6) where 𝜆2𝑛 is that eigenvalue 𝜆 of the boundary value problem 𝑟(𝐷)𝑦(𝑡)=(1)𝑟𝜆𝑞𝑄𝑞𝑥𝑄(𝑡),𝑦(𝑡)=𝑝𝑟𝑥(𝐷)𝑥(𝑡),(𝑗)(0)=𝑥(𝑗)(2𝜋),𝑦(𝑗)(0)=𝑦(𝑗)(2𝜋),𝑗=0,1,,𝑛1,(1.7) 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 Ker𝑟𝑥(𝐷)=()𝐶𝑟(𝕋)𝑟(𝐷)𝑥(𝑡)0.(2.1) 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 [2]. 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 SP2𝑘𝑝,𝑞,𝑟=(𝑥(),𝜆)SP𝑝,𝑞,𝑟𝑆𝑐(𝑥())=2𝑘.(2.2)

Let ̂𝑥2𝑛() be the solution of the extremal problem as follows: 0𝜋/2𝑛||||𝑋(𝑡)𝑞𝑑𝑡sup,0𝜋/2𝑛||𝑟||(𝐷)𝑋(𝑡)𝑝𝑥𝑑𝑡1,(𝑘)(𝜋/2𝑛)+(1)𝑘+1(𝜋/2𝑛)2=0,𝑘=0,1,,𝑛1(2.3) and the function 𝑥2𝑛() is such that 𝑥2𝑛(𝑡)=𝑥2𝑛(𝑡𝜋/𝑛) for all 𝑡𝕋𝑥2𝑛(𝑡)=̂𝑥2𝑛𝜋(𝑡),0𝑡,2𝑛̂𝑥2𝑛𝜋𝑛,𝜋𝑡𝜋2𝑛<𝑡𝑛.(2.4) 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 [6], 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 𝐺𝑟1(𝑡)=2𝜋𝑘Λ𝑒𝑖𝑘𝑡𝑝𝑟(𝑖𝑘),(2.5) where Λ={𝑘𝑝𝑟(𝑖𝑘)=0} and 𝑖 is the imaginary unit.

The 2𝜋-periodic 𝐺-splines are defined as elements of the linear space 𝑆𝑄2𝑛,𝐺𝑟𝜑=span1(𝑡),,𝜑𝜇(𝑡),𝐺𝑟𝑡𝜏1,,𝐺𝑟𝑡𝜏2𝑛.(2.6) As was proved in [7], if 𝑛𝑁, then dim 𝑆(𝑄2𝑛,𝐺𝑟)=2𝑛.

We assume (shifting 𝑥() if necessary) that 𝑟(𝐷)̂𝑥2𝑛() is positive on (𝜋,𝜋+𝜋/𝑛). Let 𝐿2𝑛=𝐿2𝑛(𝑟,𝑝,𝑞) denote the space of functions of the form 𝑥(𝑡)=𝜇𝑗=1𝑎𝑗𝜑𝑗1(𝑡)+𝜋𝕋𝐺𝑟(𝑡𝜏)2𝑛𝑖=1𝑏𝑖𝑦𝑖(𝜏)𝑑𝜏,(2.7) where 𝑎1,,𝑎𝜇,𝑏1,,𝑏2𝑛, 2𝑛𝑖=1𝑏𝑖=0, 𝑦𝑖()=𝜒𝑖()𝑟(𝐷)𝑥2𝑛((𝑖1)𝜋/𝑛), and 𝜒𝑖() is the characteristic function of the interval Δ𝑖=[𝜋+(𝑖1)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 [1]. We reformulate the definition for a linear operator 𝑃 mapping 𝑋 to 𝑌.

Definition 2.1 (see [2, page 149]). Let 𝑃𝐿(𝑋,𝑌). Then the Bernstein 𝑛-width is defined by 𝑏𝑛(𝑃(𝑋),𝑌)=sup𝑋𝑛+1inf𝑃𝑥𝑋𝑛+1𝑃𝑥0𝑃𝑥𝑌𝑥𝑋,(2.8) where 𝑋𝑛+1 is any subspace of span {𝑃𝑥𝑥𝑋} of dimension 𝑛+1.

2.1. Lower Estimate of Bernstein 𝑛-Width

Consider the extremal problem 𝑥()𝑞𝑞𝑟(𝐷)𝑥()𝑝𝑝inf,𝑥()𝐿2𝑛(2.9) and denote the value of this problem by 𝛼𝑞. Let us show that 𝛼𝜆𝑛; this will imply the desired lower bound for 𝑏2𝑛1. Let 𝑥()𝐿2𝑛, then𝑟(𝐷)𝑥()𝑝𝑝=2𝑛𝑖=1Δ𝑖|||||2𝑛𝑖=1𝑏𝑖𝑦𝑖|||||(𝑡)𝑝𝑑𝑡=2𝑛𝑖=1Δ𝑖||𝑏𝑖||𝑝||𝑟(𝐷)𝑥𝑛||(𝑡)𝑝1𝑑𝑡=2𝑛2𝑛𝑖=1||𝑏𝑖||𝑝(2.10) and by setting 𝑧𝑖(1)=𝜋𝕋𝐺𝑟(𝜏)𝑦𝑖(𝜏)𝑑𝜏,𝑖=1,2,,2𝑛,(2.11)

we reduce problem (2.9) to the form 𝜇𝑗=1𝑎𝑗𝜑𝑗()+2𝑛𝑖=1𝑏𝑖𝑧𝑖()𝑞𝑞1/2𝑛2𝑛𝑖=1||𝑏𝑖||𝑝inf,𝑎1,,𝑎𝜇,𝑏1,,𝑏2𝑛.(2.12)

This is a smooth finite-dimensional problem. It has a solution (𝑎1,𝑎𝜇,𝑏1,,𝑏2𝑛) and, (𝑏1,,𝑏2𝑛0). 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 (𝑎1,,𝑎𝜇,𝑏1,,𝑏2𝑛) are equal to zero. This leads to the relations𝕋𝜑𝑗(𝑄𝑡)𝑞𝑥(𝑡)𝑑𝑡=0,𝑗=1,,𝜇,(2.13)𝕋𝑧𝑖𝑄(𝑡)𝑞𝑥1(𝑡)𝑑𝑡=2𝑛𝑥()𝑞𝑞𝑟(𝐷)𝑥()𝑝𝑝𝑄𝑝𝑏𝑖,𝑖=1,,2𝑛,(2.14) where 𝑥()=𝜇𝑗=1𝑎𝑗𝜑𝑗(𝑡)+2𝑛𝑖=1𝑏𝑖𝑧𝑖().

We remark that 𝑔(𝑎1,,𝑎𝜇,𝑏1,,𝑏2𝑛)=𝑔(𝑑𝑎1,,𝑑𝑎𝜇,𝑑𝑏1,,𝑑𝑏2𝑛) for any 𝑑0, and hence the vector (𝑑𝑎1,,𝑑𝑎𝜇,𝑑𝑏1,,𝑑𝑏2𝑛) is also a solution of (2.12). Thus, it can be assumed that |𝑏𝑖|1, 𝑖=1,,2𝑛 and 𝑏𝑖0=(1)𝑖0+1 for some 𝑖0, 1𝑖02𝑛.

Let ̃𝑥2𝑛(𝑡)=𝜇𝑗=1𝑎𝑗𝜑𝑗(𝑡)+2𝑛𝑖=1(1)𝑖+1𝑧𝑖(𝑡)(2.15) and ̃𝑥2𝑛 satisfies (1.7). Let 𝑎=(𝑎1,,𝑎2𝑛) and let 𝑏=(1,1,,1,1)2𝑛. It follows from the definitions of ̃𝑥2𝑛() and 𝑥() that 𝑟(𝐷)̃𝑥2𝑛(𝑡)𝑟(𝐷)𝑥(𝑡)=2𝑛𝑖=1𝑖𝑖0(1)𝑖+1𝑏𝑖𝜒𝑖(𝑡)𝑟(𝐷)𝑥2𝑛𝑡(𝑖1)𝜋𝑛,(2.16) and hence 𝑆𝑐(𝑟(𝐷)̃𝑥2𝑛(),𝑟(𝐷)𝑥()) has at most 2𝑛2 sign changes. Then, by Rolle's theorem, 𝑆𝑐(𝑟(𝐷)̃𝑥2𝑛()𝑟(𝐷)𝑥())2𝑛2. For any 𝑎,𝑏, sign(𝑎+𝑏)=sign(𝑄𝑝𝑎+𝑄𝑝𝑏); therefore 𝑆𝑐𝑄𝑞̃𝑥2𝑛𝑄()𝑞𝑥()2𝑛2.(2.17)

In addition, since ̃𝑥2𝑛 is 2𝜋-periodic solution of the linear differential equation, 𝑟(𝐷)𝑦(𝑡)=(1)𝑟𝜆𝑞(𝑄𝑞𝑥)(𝑡) and 𝜑𝑗(𝑡)Ker𝑟(𝐷). Then, by [8, page 94], we have𝕋𝜑𝑗(𝑄𝑡)𝑞(̃𝑥𝑡)𝑑𝑡=0,𝑗=1,,𝜇.(2.18)

If we now multiply both sides of (2.15) by (𝑄𝑞̃𝑥2𝑛)(𝑡) and integrate over the interval Δ𝑖, 1𝑖2𝑛, we getΔ𝑖𝑧𝑖𝑄(𝑡)𝑞̃𝑥2𝑛(𝑡)𝑑𝑡=(1)𝑖+1Δ𝑖||̃𝑥2𝑛||(𝑡)𝑞𝑑𝑡=(1)𝑖+1𝜆𝑞2𝑛2𝑛,(2.19) due to 𝕋𝑧𝑖(𝑡)(𝑄𝑞̃𝑥2𝑛)(𝑡)𝑑𝑡=Δ𝑖𝑧𝑖(𝑡)(𝑄𝑞̃𝑥2𝑛)(𝑡)𝑑𝑡. Therefore, we have𝕋𝑧𝑖𝑄(𝑡)𝑞̃𝑥2𝑛(𝑡)𝑑𝑡=(1)𝑖+1𝜆𝑞2𝑛2𝑛,𝑖=1,,2𝑛.(2.20) Changing the order of integration and using (2.14) and (2.20), we get that Δ𝑖𝑟(𝐷)𝑥2𝑛𝑡(𝑖1)𝜋𝑛1𝜋𝕋𝐺𝑟𝑄(𝑡𝜏)𝑞̃𝑥2𝑛𝑄(𝜏)𝑞𝑥=(𝜏)𝑑𝜏𝑑𝑡𝕋𝑧𝑖𝑄(𝑡)𝑞̃𝑥2𝑛𝑄(𝑡)𝑞𝑥(𝑡)𝑑𝑡=(1)𝑟2𝑛(1)𝑖+1𝜆𝑞2𝑛𝑥()𝑞𝑞𝑟(𝐷)𝑥()𝑝𝑝𝑄𝑝𝑏𝑖=(1)𝑟(2𝑛1)𝑖+1𝜆𝑞2𝑛𝛼𝑞𝑟(𝐷)𝑥()𝑞𝑝𝑄𝑝𝑏𝑖,𝑖=1,,2𝑛.(2.21) Denote by 𝑓() the factor multiply 𝑟(𝐷)𝑥2𝑛(𝑡(𝑖1)𝜋/𝑛) in the integral in the left-hand side of this equality. Since (𝑟(𝐷)𝑥)()𝑝1 and hence (𝑟(𝐷)𝑥)()𝑝𝑞𝑝1 for 𝑝𝑞, if we assume that 𝜆2𝑛>𝛼, then we arrive at the relationssignΔ𝑖𝑟(𝐷)𝑥2𝑛𝑡(𝑖1)𝜋𝑛𝑓()𝑑𝑡=(1)𝑟+𝑖+1,𝑖=1,,2𝑛.(2.22)

Suppose for definiteness that 𝑟(𝐷)𝑥2𝑛(𝑡(𝑖1)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())2𝑛1. But 𝑓() is periodic, and hence 𝑆𝑐(𝑓())2𝑛; therefore, 𝑆𝑐(𝑟(𝐷)𝑓())2𝑛. Further, 𝑟(𝐷)𝑓()=(𝑄𝑞̃𝑥2𝑛)(𝑡)(𝑄𝑞𝑥)(𝑡), that is, 𝑆𝑐((𝑄𝑞̃𝑥2𝑛)(𝑡)(𝑄𝑞𝑥)(𝑡))2𝑛.

We have arrived at a contradiction to (2.17), and hence 𝜆2𝑛𝛼. Thus 𝑏2𝑛1(𝑊𝑝(𝑟);𝐿𝑞)𝜆2𝑛.

2.2. Upper Estimate of Bernstein 𝑛-Width

Assume the contrary: 𝑏2𝑛1(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()𝐿2𝑛(𝑥)𝑞𝑟(𝐷)𝑥()𝑝𝛾>𝜆2𝑛.(2.23) Let us assign a vector 𝑐2𝑛 to each function 𝑥()𝐿2𝑛 by the following rule: 𝑐𝑥()𝑐=1,,𝑐2𝑛2𝑛,where𝑥()=2𝑛𝑗=1𝑐𝑗𝑓𝑗().(2.24) Then inequality (2.23) acquires the form min𝑐2𝑛{0}2𝑛𝑗=1𝑐𝑗𝑓𝑗()𝑞2𝑛𝑗=1𝑐𝑗𝑟(𝐷)𝑓𝑗()𝑝𝛾>𝜆2𝑛.(2.25) Let 𝑐0=0. Consider the sphere 𝑆2𝑛1 in the space 2𝑛 with radius 2𝜋, that is, 𝑆2𝑛1𝑐=𝑐𝑐=1,,𝑐2𝑛2𝑛,𝑐=2𝑛𝑗=1||𝑐𝑗||=2𝜋.(2.26) To every vector 𝑐2𝑛 we assign function 𝑢(𝑡,𝑐) defined by 𝑢(𝑡,𝑐)=(2𝜋)1/𝑝sign𝑐𝑘𝑡,for𝑡𝑘1,𝑡𝑘,𝑘=1,,2𝑛,0,for𝑡=𝑡𝑘,𝑘=1,,2𝑛1,(2.27) where 𝑡0=0, 𝑡𝑘=𝑘𝑖=1|𝑐𝑖|, 𝑘=1,,2𝑛, and the extended 2𝜋-periodically onto .

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 𝑟(𝐷)𝑥0=𝑢; then the periodic functions {𝑥𝑘(𝑡,𝑐)}𝑘+ are successively determined from the differential equations𝑟(𝐷)𝑥𝑘𝑄(𝑡)=𝑝𝑦𝑘(𝑡),𝑟(𝐷)𝑦𝑘(𝑡)=(1)𝑟𝜇𝑞𝑘1𝑄𝑞𝑥𝑘1(𝑡),(2.28) where 𝑝=𝑝/(𝑝1), and the constants {𝜇𝑘𝑘=0,} are uniquely determined by the conditions 𝑟(𝐷)𝑥𝑘𝑝𝑄=1,𝑞𝑥𝑘(𝑡)Ker𝑟𝑄(𝐷),𝑝𝑦𝑘(𝑡)Ker𝑟(𝐷).(2.29)

By analogy with the reasoning in [5], we can prove the following assertions.(i)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).(ii)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.(iii)For any 𝑘 there exists a ̂𝑐𝑆2𝑛1 such that 𝑥𝑘(,̂𝑐) has at least 2𝑛 zeros (𝑍𝑐(𝑥𝑘(,̂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 𝑆𝑐(𝑥(,̂𝑐)=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 [10], which states that there exists a ̂𝑐𝑆2𝑛1 such that 𝑍𝑐(𝑥𝑘(,̂𝑐))2𝑛1, but since the function 𝑥𝑘(,̂𝑐) is periodic, we actually have 𝑍𝑐(𝑥𝑘(,̂𝑐))2𝑛. Finally, in item (iv), by (ii) and (iii), 𝑍𝑐(𝑥(,̂𝑐))2𝑛. In view of 𝑥(,̂𝑐) zeros are simple; therefore, 𝑆𝑐(𝑥(,̂𝑐))2𝑛.

Note that [8] 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(̂𝑐)=𝜆2𝑁𝜆2𝑛.(2.30) Therefore, by virtue of items (i), (ii), and (2.30), we obtain min𝑐2𝑛{0}2𝑛𝑗=1𝑐𝑗𝑓𝑗()𝑞2𝑛𝑗=1𝑐𝑗𝑟(𝐷)𝑓𝑗()𝑝2𝑛𝑗=1̂𝑐𝑗𝑓𝑗()𝑞2𝑛𝑗=1̂𝑐𝑗𝑟(𝐷)𝑓𝑗()𝑝𝑥𝑘(,̂𝑐)𝑞𝑟(𝐷)𝑥𝑘(,̂𝑐)𝑝𝜆(̂𝑐)𝜆2𝑛,(2.31) which contradicts (2.25). Hence 𝑏2𝑛1(𝑊𝑝(𝑟);𝐿𝑞)𝜆2𝑛. Thus, the upper bound is proved. This completes the proof of the theorem.


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).