Abstract
We consider the classes of periodic functions with formal self-adjoint linear differential operators , 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 in the space for .
1. Introduction and Main Result
Let , , , , and be the sets of all complex numbers, real numbers, integers, nonnegative integers, and positive integers, respectively.
Let be the unit circle realized as the interval with the points 0 and identified, and as usual, let be the classical Lebesgue integral space of -periodic real-valued functions with the usual norm , . Denote by the Sobolev space of functions on such that the st derivative is absolutely continuous on and , . The corresponding Sobolev class is the set
Tikhomirov [1] introduced the notion of Bernstein width of a centrally symmetric set in a normed space . It is defined by the formula where is the unit ball of and the outer supremum is taken over all subspaces such that dim , .
In particular, Tikhomirov posed the problem of finding the exact value of , where and , , . He also obtained the first results [1] for and . Pinkus [2] found , where . Later, Magaril-Il'yaev [3] obtained the exact value of for . The latest contribution to these fields is due to Buslaev et al. [4] who found the exact values of for all .
Let be an arbitrary linear differential operator of order with constant real coefficients . Denote by the characteristic polynomial of . The linear differential operator will be called formal self-adjoint if for each .
We define the function classes as follows: where .
In this paper, we consider some classes of periodic functions with formal self-adjoint linear differential operators , 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 in the space for . The results of Buslaev et al. [4] are extended to the classes (1.4) defined by differential operators (1.3).
We define to be the nonlinear transformation
The main result of this paper is the following.
Theorem 1.1. Let be an arbitrary formal self-adjoint linear differential operator given by (1.3), and , . Then where is that eigenvalue of the boundary value problem for which the corresponding eigenfunction has only simple zeros on and is normalized by the condition .
2. Proof of the Theorem
First we introduce some notations and formulate auxiliary statements.
Let be an arbitrary linear differential operator (1.3). Denote the -periodic kernel of by Let be the dimension of and an arbitrary basis in .
denotes the number of zeros of on a period, counting multiplicity, and is the cyclic sign change count for a piecewise continuous, -Periodic function [2]. Following, is called the spectral pair of (1.7) if the function is normalized by the condition . The set of all spectral pairs is denoted by . Define the spectral classes as
Let be the solution of the extremal problem as follows: and the function is such that for all Let us extend periodically the function 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 belonging to . Furthermore, by [6], which any other function from differs from only in the sign and in a shift of its argument, and there exists a number such that for every , all zeros of are simple, equidistant with a step equal to , and . We denote the set of zeros (= sign variations) of on the period by . Let where and is the imaginary unit.
The -periodic -splines are defined as elements of the linear space As was proved in [7], if , then dim .
We assume (shifting if necessary) that is positive on . Let denote the space of functions of the form where , , , and is the characteristic function of the interval , . Obviously, dim and .
Let us now consider exact estimate of Bernstein -width. This was introduced in [1]. We reformulate the definition for a linear operator mapping to .
Definition 2.1 (see [2, page 149]). Let . Then the Bernstein -width is defined by where is any subspace of span of dimension .
2.1. Lower Estimate of Bernstein -Width
Consider the extremal problem and denote the value of this problem by . Let us show that ; this will imply the desired lower bound for . Let , then and by setting
we reduce problem (2.9) to the form
This is a smooth finite-dimensional problem. It has a solution and, . According to the Lagrange multiplier rule, there exists a such that the derivatives of the function (where is the function being minimized in (2.12)) with respect to at the point are equal to zero. This leads to the relations where .
We remark that for any , and hence the vector is also a solution of (2.12). Thus, it can be assumed that , and for some , .
Let and satisfies (1.7). Let and let . It follows from the definitions of and that and hence has at most sign changes. Then, by Rolle's theorem, . For any , ; therefore
In addition, since is -periodic solution of the linear differential equation, and . Then, by [8, page 94], we have
If we now multiply both sides of (2.15) by and integrate over the interval , , we get due to . Therefore, we have Changing the order of integration and using (2.14) and (2.20), we get that Denote by the factor multiply in the integral in the left-hand side of this equality. Since and hence for , if we assume that , then we arrive at the relations
Suppose for definiteness that interior to , . Then it follows from (2.22) that there are points such that , , that is, . But is periodic, and hence ; therefore, . Further, , that is, .
We have arrived at a contradiction to (2.17), and hence . Thus .
2.2. Upper Estimate of Bernstein -Width
Assume the contrary: , . Then, by definition, there exists a linearly independent system of functions and number such that , or equivalently, Let us assign a vector to each function by the following rule: Then inequality (2.23) acquires the form Let . Consider the sphere in the space with radius , that is, To every vector we assign function defined by where , , , and the extended -periodically onto .
An analog of the Buslaev iteration process [5] is constructed in the following way: the function is found as a periodic solution of the linear differential equation ; then the periodic functions are successively determined from the differential equations where , and the constants are uniquely determined by the conditions
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 are monotone nondecreasing and converge to an eigenvalue of the problem (1.7).(ii)The sequence has a subsequence that is convergent to an eigenfunction of the problem (1.7), with .(iii)For any there exists a such that has at least zeros on .(iv)In the set of spectral pairs , there exists a pair such that .
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 such that , but since the function is periodic, we actually have . Finally, in item (iv), by (ii) and (iii), . In view of zeros are simple; therefore, .
Note that [8] the linear differential equation has a -periodic solution if and only if , where and is an integrable -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 is monotone decreasing for ; it follows that Therefore, by virtue of items (i), (ii), and (2.30), we obtain which contradicts (2.25). Hence . 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).