Volume 2008, Article ID 894529,8pages 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,[email protected]
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
WpLr, 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 Bernsteinn-width of the classesWpLrin theLpfor 1< p <∞.
Copyrightq2008 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
LetC,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, letLq : Lq0,2π be the classical Lebesgue integral space of 2π-periodic real-valued functions with the usual norm·q, 1≤q ≤
∞. Denote byWr
p the Sobolev space of functions x·onT such that ther−1st derivative
xr−1·is absolutely continuous onTandxr·∈Lp, r ∈N. The corresponding Sobolev class
is the set
Wpr :Wpr :xr·p ≤1
. 1.1
Tikhomirov1introduced the notion of Bernstein width of a centrally symmetric setC
in a normed spaceX. It is defined by the following formula:
bnC, X:sup L
whereBXis the unit ball ofXand the outer supremum is taken over all subspacesL⊂Xsuch that dimL≥n1, n∈N.
In particular, Tikhomirov posed the problem of finding the exact value ofbnC;X, where
C Wr
p andX Lq, 1 ≤ p, q ≤ ∞. He also obtained the first results1forp q ∞ and
n2k−1. Pinkus2foundb2n−1Wpr;Lq, wherepq1. Later, Magaril-Il’yaev3obtained
the exact value ofb2n−1Wpr;Lp,for 1 < p < ∞. The latest contribution to this fields is due to
Buslaev et al.4who found the exact values ofb2n−1Wpr;Lqfor all 1< p≤q <∞.
Let
LrD Drar−1Dr−1· · ·a1Da0, D d
dt, 1.3
be an arbitrary linear differential operator of order r with constant real coefficients
a0, a1, . . . , ar−1. Denote by pr the characteristic polynomial of LrD. The linear differential
operatorLrDwill be called formal self-adjoint ifpr−t −1rprt,for eacht∈C.
We define the function classesWpLras follows:
Wp
Lr
x·:xr−1∈AC2π,LrDx·p ≤1
, 1.4
where 1≤p≤ ∞.
In this paper, we will determine the exact values of Bernsteinn-width of some classes of periodic functions with formal self-adjoint linear differential operatorsWpLr, which include
the classical Sobolev class as its special case.
We defineQpto be the nonlinear transformation
Qpf
t:ftp−1signft. 1.5
The maim result of this paper is the following.
Theorem 1.1. Assume that1< p <∞. LetLrDbe an arbitrary formal self-adjoint linear differential
operators given by1.3. Then, there exists a numberN ∈Nsuch that for everyn≥N:
b2n−1
WpLr
;Lp λ2n:λ2n
p, p,Lr
, 1.6
whereλ2nis that eigenvalueλof the boundary value problem
LrDyt −1rλ−p
Qpx
t,
yt QpLrDx
t,
xj0 xj2π, yj0 yj2π, j0,1, . . . , n−1,
1.7
for which the corresponding eigenfunctionx· x2n·has only2nsimple zeros onTand is normalized
2. Proof of the theorem
First we introduce some notations and formulate auxiliary statements.
LetLrDbe an arbitrary linear differential operator1.3. Denote the 2π-periodic kernel
ofLrDby
KerLrD
x·∈CrT:LrDxt≡0
. 2.1
Letμ0≤μ≤rbe the dimension of KerLrDand{ϕi, . . . , ϕμ}an arbitrary basis in KerLrD.
Zcfdenotes the number of zeros off in a period, counting multiplicity, andScfis
the cyclic sign change count for a piecewise continuous, 2π-periodic functionf2. Following,
x·, λis called the spectral pair of 1.7if the functionx·is normalized by the condition LrDx·p 1. The set of all spectral pairs is denoted by SPp, p,Lr. Define the spectral
classes SP2kp, p,Lras
SP2kp, p,Lr
x·, λ∈SPp, p,Lr:Sc
x·2k. 2.2
Letx2n·denotes the solution of the extremal problem as follows:
π/2n
0
|Xt|pdt−→sup,
π/2n
0
|LrDXt|pdt≤1,
xk
π
2n −1
k1π
2n
/2
0, k0,1, . . . , n−1,
2.3
and the functionx2n·is such thatx2nt −x2nt−π/nfor allt∈T:
x2nt:
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩
x2nt, 0≤t≤ π
2n,
x2n
π n −t
, π
2n < t≤ π
n.
2.4
Let us extend periodically the functionx2ntontoR, and normalize the obtained function as
it is required in the definition of spectral pairs. From what has been done above, we get a functionx2ntbelongs to SP2np, p,Lr. Furthermore, by5, which any other function from
SP2np, p,Lrdiffers fromx2n·only in the sign and in a shift of its argument, and there exists a
numberN ∈Nsuch that for everyn≥N, all zeros ofx2n·are simple, equidistant with a step
equal toπ/n, andScx2n ScLrDx2n 2n. We denote the set of zerossign variations
ofLrDx2non the period byQ2n τ1, . . . , τ2n. Let
Grt 1
2π
k/∈Λ eikt
prik, 2.5
The 2π-periodicG-splines are defined as elements of the linear space
SQ2n, Gr
spanϕ1t, . . . , ϕμt, Gr
t−τ1
, . . . , Gr
t−τ2n
. 2.6
As was proved in6, ifn≥N, then dimSQ2n, Gr 2n.
We assumeshiftingx·if necessarythatLrDx2n·is positive on−π, ππ/n. Let
L2n:L2nr, p, pdenote the space of functions of the form
xt
μ
j1
ajϕjt 1
π TGrt−τ
2n
i1
biyiτ
dτ, 2.7
where a1, . . . , aμ, b1, . . . , b2n ∈ R,
2n
i1bi 0, yi· χi·LrDx2n· −i−1π/n, andχi·
is the characteristic function of the intervalΔi : −π i−1π/n,−π iπ/n, 1 ≤ i ≤ 2n.
Obviously, dimL2n2nandL2n ⊂WpLr.
Let us now consider exact estimate of Bernsteinn-width. This was introduced in1. We reformulate the definition for a linear operatorP mappingXtoY.
Definition 2.1see2, page 149. LetP ∈LX, Y. Then the Bernsteinn-width is defined by
bnPX, Y sup Xn1
inf
P x∈Xn1 P x /0
P xY
xX , 2.8
whereXn1is any subspace of span{P x:x∈X}of dimension≥n1.
2.1. Lower estimate of Bernsteinn-width
Consider the extremal problem
x·pp
LrDx·pp
−→inf, x·∈L2n, 2.9
and denote the value of this problem byαp. Let us show thatα≥λ
n, this will imply the desired
lower bound forb2n−1. Letx·∈L2n, then
LrDx·pp
2n
i1 Δi
2n
i1
biyit
p
dt
2n
i1 Δi
|bi|p|LrDxnt|pdt 1
2n
2n
i1
|bi|p, 2.10
and by setting
zi·: 1
π TGr· −τyiτdτ, i1,2, . . . ,2n, 2.11
we reduce problem2.9to the form
μ
j1ajϕj·
2n
i1bizi·pp
1/2n2in1|bi|p
This is a smooth finite-dimensional problem. It has a solutiona1, . . . aμ, b1, . . . , b2n, and,
moreover, b1, . . . , b2n/0. According to the Lagrange multiplier rule, there exists a η ∈ R
such that the derivatives of the function a1, . . . , aμ, b1, . . . , b2n→ga1, . . . , aμ, b1, . . . , b2n
ηb1 b2 · · · b2n where g· is the function being minimized in2.12 with respect to
a1, . . . , aμ, b1, . . . , b2n at the point a1, . . . aμ, b1, . . . , b2n are equal to zero. This leads to the
relations
Tϕjt
Qpx
tdt0, j 1, . . . , μ, 2.13
Tzit
Qpx
tdt α
p
2nQpbi, i1, . . . ,2n, 2.14
wherex· μj1ajϕjt
2n
i1bizi·.
We remark that ga1, . . . , aμ, b1, . . . , b2n gda1, . . . , daμ, db1, . . . , db2n for any d /0,
and hence the vectord a1, . . . , d aμ, d b1, . . . , d b2nis also a solution of 2.12. Thus, it can be
assumed that|bi| ≤1, i1, . . . ,2n, andbi0 −1i01for somei0, 1≤i0≤2n.
Let
x2nt μ
j1
ajϕjt
2n
i1
−1i1z
it, 2.15
andx2nsatisfies1.7. Leta a1, . . . , a2nandb 1,−1, . . . ,1,−1∈R2n. It follows from the
definitions ofx2n·andx·that
LrDx2nt− LrDxt
2n
i1
i /i0
−1i1−b
i
χitLrDx2n
t−i−1π n
, 2.16
and henceScLrDx2n·,LrDx·has at most 2n−2 sign changes. Then, by Rolle’s theorem,
ScLrDx2n·− LrDx·≤2n−2. For anya, b∈R,signab signQpaQpb, therefore
Sc
Qpx2n
·−Qpx·
≤2n−2. 2.17
In addition, since x2n is 2π-periodic solution of the linear differential equation
LrDyt −1rλ−pQpxt, andϕjt∈KerLrD. Then, by7, page 94, we have
Tϕjt
Qpx
tdt0, j 1, . . . , μ. 2.18
If we now multiply both sides of 2.15by Qpx2nt, and integrate over the interval
Δi, 1≤i≤2n, we get
Δizit
Qpx2n
tdt −1i1
Δi|x2nt|
pdt −1i1λ
p
2n
2n. 2.19
Due toTzit
Qpx2n
tdtΔizit
Qpx2n
tdt. Therefore, we have
Tzit
Qpx2n
tdt −1i1λ
p
2n
Changing the order of integration and using2.14and2.20, we get that
ΔiLrDx2n
t− i−1π n
1
π TGrt−τ
Qpx2n
τ−Qpx
τdτ
dt
Tzit
Qpx2n
t−Qpxt
dt 1
2n
−1i1λp
2n−αpQpbi
.
2.21
Denote byf·the factor multiplyLrDx2nt−i−1π/nin the integral in the left-hand side
of this equality. If we assume thatλ2n> α, then we arrive at the relations
sign
ΔiLrDx2n
t−i−1π n
f·dt −1i1, i1, . . . ,2n. 2.22
Suppose for definiteness thatLrDx2nt−i−1π/n> 0 interior toΔi, i 1, . . . ,2n.
Then it follows from 2.22 that there are points ti ∈ Δi such that signfti −1i1, i
1, . . . ,2n, that is, Scf· ≥ 2n−1. But f· is periodic, and hence Scf· ≥ 2n, therefore,
ScLrDf· ≥ 2n. Further, LrDf·
Qpx2n
t− Qpxt, that is, ScQpx2nt −
Qpxt
≥2n.
We have arrived at a contradiction to2.17, and henceλ2n≤α. Thusb2n−1WpLr;Lp≥
λ2n.
2.2. Upper estimate of Bernsteinn-width
Assume the contrary:b2n−1WpLr;Lp> λ2n,1 < p <∞. Then, by definition, there exists a
linearly independent system of 2nfunctionsL2n :span
f1, . . . , f2n
⊂Lp and numberγ > λ
2n
such thatL2n∩γSLp⊆ LrD, or equivalently,
min
x·∈L2n
x·p
LrDx·p ≥
γ > λ2n. 2.23
Let us assign a vectorc∈R2nto each functionx·∈L
2nby the following rule:
x·−→c c1, . . . , c2n∈R2n, wherex·
2n
j1
cjfj·. 2.24
Then2.23acquires the form
min
c∈R2n\{0}
2n
j1cjfj·p
2n
j1cjLrDfj·p
≥γ > λ2n. 2.25
Letc00. Consider the sphereS2n−1in the spaceR2nwith radius 2π, that is,
S2n−1:
c:c c1, . . . , c2n∈R2n, c
2n
j1
|cj|2π
To every vectorc∈R2nwe assign functionut, cdefined by
ut, c
⎧ ⎨ ⎩
2π−1/psigncj, fort∈
tk−1, tk
, k 1, . . . ,2n,
0, forttk, k1, . . . ,2n−1,
2.27
wheret00, tk
k
i1|ci|, k1, . . . ,2n, and the extended 2π-periodically ontoR.
An analog of the Buslaev iteration process8is constructed in the following way: the function xt, cis found as a periodic solution of the linear differential equation LrDx0
u, then the periodic functions{xkt, c}k∈N are successively determined from the differential
equations
LrDxkt
Qpyk
t,
LrDykt −1rμ−kp−1
Qpxk−1
t, 2.28
where p p/p−1, and the constants {μk : k 0, . . . ,} are uniquely determined by the
conditions
LrDxkp 1,
Qpxk
t⊥KerLrD,
Qpyk
t⊥KerLrD. 2.29
By analogy with the reasoning in8, we can prove the following assertions:
ithe iteration procedure 2.28-2.29 is well de fined, the sequences {μk}k∈N is
monotone nondecreasing and converge to an eigenvalueλc>0 of the problem1.7,
iithe sequence {xk·, c}k∈N has a subsequence that is convergent to an eigenfunction
x·, cof the problem1.7, withλc x·, cp,
iiifor any k ∈ N there exists a c ∈ S2n−1 such that x
k·,c has at least 2n zeros
Zcxk·,c≥2nonT,
ivin the set of spectral pairs λc, x·, c, there exists a pair λc, x·,c such that
Scx·,c 2N≥2n.
Items iand 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 ∈ S2n−1 such that
Zcxk·,c≥2n−1, but since the functionxk·,cis periodic, we actually haveZcxk·,c≥2n.
Finally, itemiv, byiiand iii, which Zcx·,c ≥ 2n. In view ofx·,czeros are simple,
therefore,Scx·,c≥2n.
Since spectral pairs of 1.7 are unique and the Kolmogorov widthd2nWpLr;Lq
λ2np, q,Lrforp≥q5, whenn≥N, it follows that
λc λ2N d2N
Wp
Lr
;Lp≤d2n
Wp
Lr
;Lpλ2n. 2.30
Therefore, by virtue of itemsi,ii, and2.30, we obtain
min
c∈R2n\{0}
2n
j1cjfj·p
2n
j1cjLrDfj·p
≤
2n
j1cjfj·p
2n
j1cjLrDfj·p
≤ Lxk·,cp
rDxk·,cp ≤
λc λ2N ≤λ2n,
2.31
which contradicts 2.25. Hence b2n−1WpLr;Lp ≤ λ2n. Thus, the upper bound is proved.
Acknowledgments
Project was supported by the Natural Science Foundation of ChinaGrant no. 10671019and Scientific Research Fund of Zhejiang Provincial Education DepartmentGrant no. 20070509.
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