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Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2008, Article ID 894529, 8 pages
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,
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
W
p
L
r
, 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 Bernstein n-width of the classes W
p
L
r
in
the L
p
for 1 <p<∞.
Copyright q 2008 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
Let
C, 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, let L
q
: L
q
0, 2π be the classical
Lebesgue integral space of 2π-periodic real-valued functions with the usual norm ·
q
, 1 ≤ q ≤
∞.Denoteby
W
r
p
the Sobolev space of functions x· on T such that the r − 1st derivative
x
r−1
· is absolutely continuous on T and x
r
· ∈ L
p
,r∈ N. The corresponding Sobolev class
is the set
W
r
p
:
W
r
p
: x
r
·
p
≤ 1
. 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 following formula:
b
n
C, X : sup
L
sup{λ ≥ 0:L ∩ λBX ⊂ C}, 1.2
2 Journal of Inequalities and Applications
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 b
n
C; X,where
C W
r
p
and X L
q
, 1 ≤ p, q ≤∞. He also obtained the first results 1 for p q ∞ and
n 2k − 1. Pinkus 2 found b
2n−1
W
r
p
; L
q
,wherep q 1. Later, Magaril-Il’yaev 3 obtained
the exact value of b
2n−1
W
r
p
; L
p
, for 1 <p<∞. The latest contribution to this fields is due to
Buslaevetal.4 who found the exact values of b
2n−1
W
r
p
; L
q
for all 1 <p≤ q<∞.
Let
L
r
DD
r
a
r−1
D
r−1
··· a
1
D a
0
,D
d
dt
, 1.3
be an arbitrary linear differential operator of order r with constant real coefficients
a
0
,a
1
, ,a
r−1
.Denotebyp
r
the characteristic polynomial of L
r
D. The linear differential
operator L
r
D will be called formal self-adjoint if p
r
−t−1
r
p
r
t, for each t ∈ C.
We define the function classes W
p
L
r
as follows:
W
p
L
r
x· : x
r−1
∈ AC
2π
, L
r
Dx·
p
≤ 1
, 1.4
where 1 ≤ p ≤∞.
In this paper, we will determine the exact values of Bernstein n-width of some classes of
periodic functions with formal self-adjoint linear differential operators W
p
L
r
, which include
the classical Sobolev class as its special case.
We define Q
p
to be the nonlinear transformation
Q
p
f
t :
ft
p−1
signft. 1.5
The maim result of this paper is the following.
Theorem 1.1. Assume that 1 <p<∞.LetL
r
D be an arbitrary formal self-adjoint linear differential
operators given by 1.3. Then, there exists a number N ∈
N
such that for every n ≥ N:
b
2n−1
W
p
L
r
; L
p
λ
2n
: λ
2n
p, p,L
r
, 1.6
where λ
2n
is that eigenvalue λ of the boundary value problem
L
r
Dyt−1
r
λ
−p
Q
p
x
t,
yt
Q
p
L
r
Dx
t,
x
j
0x
j
2π,y
j
0y
j
2π,j 0, 1, ,n− 1,
1.7
for which the corresponding eigenfunction x·x
2n
· has only 2n simple zeros on T and is normalized
by the condition L
r
Dx·
p
1.
Feng Guo 3
2. Proof of the theorem
First we introduce some notations and formulate auxiliary statements.
Let L
r
D be an arbitrary linear differential operator 1.3.Denotethe2π-periodic kernel
of L
r
D by
KerL
r
D
x· ∈ C
r
T : L
r
Dxt ≡ 0
. 2.1
Let μ 0 ≤ μ ≤ r be the dimension of KerL
r
D and {ϕ
i
, ,ϕ
μ
} an arbitrary basis in KerL
r
D.
Z
c
f denotes the number of zeros of f in a period, counting multiplicity, and S
c
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
L
r
Dx·
p
1. The set of all spectral pairs is denoted by SPp, p, L
r
. Define the spectral
classes SP
2k
p, p,L
r
as
SP
2k
p, p,L
r
x·,λ
∈ SP
p, p,L
r
: S
c
x·
2k
. 2.2
Let x
2n
· denotes the solution of the extremal problem as follows:
π/2n
0
|Xt|
p
dt −→ sup,
π/2n
0
|L
r
DXt|
p
dt ≤ 1,
x
k
π
2n
−1
k1
π
2n
/2
0,k 0, 1, ,n− 1,
2.3
and the function x
2n
· is such that x
2n
t−x
2n
t − π/n for all t ∈ T:
x
2n
t :
⎧
⎪
⎪
⎨
⎪
⎪
⎩
x
2n
t, 0 ≤ t ≤
π
2n
,
x
2n
π
n
− t
,
π
2n
<t≤
π
n
.
2.4
Let us extend periodically the function x
2n
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 x
2n
t belongs to SP
2n
p, p,L
r
. Furthermore, by 5, which any other function from
SP
2n
p, p,L
r
differs from x
2n
· only in the sign and in a shift of its argument, and there exists a
number N ∈
N
such that for every n ≥ N,allzerosofx
2n
· are simple, equidistant w ith a step
equal to π/n,andS
c
x
2n
S
c
L
r
Dx
2n
2n. We denote the set of zeros sign variations
of L
r
Dx
2n
on the period by Q
2n
τ
1
, ,τ
2n
.Let
G
r
t
1
2π
k
/
∈Λ
e
ikt
p
r
ik
, 2.5
where Λ{k ∈
Z : p
r
ik0} and i is the imaginary unit.
4 Journal of Inequalities and Applications
The 2π-periodic G-splines are defined as elements of the linear space
S
Q
2n
,G
r
span
ϕ
1
t, ,ϕ
μ
t,G
r
t − τ
1
, ,G
r
t − τ
2n
. 2.6
As was proved in 6,ifn ≥ N,thendimSQ
2n
,G
r
2n.
We assume shifting x· if necessary that L
r
Dx
2n
· is positive on −π, π π/n.Let
L
2n
: L
2n
r, p, p denote the space of functions of the form
xt
μ
j1
a
j
ϕ
j
t
1
π
T
G
r
t − τ
2n
i1
b
i
y
i
τ
dτ, 2.7
where a
1
, ,a
μ
,b
1
, ,b
2n
∈ R,
2n
i1
b
i
0,y
i
·χ
i
·L
r
Dx
2n
·−i − 1π/n,andχ
i
·
is the characteristic function of the interval Δ
i
:−π i − 1π/n,−π iπ/n, 1 ≤ i ≤ 2n.
Obviously, dim L
2n
2n and L
2n
⊂ W
p
L
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.LetP ∈ LX, Y . Then the Bernstein n-width is defined by
b
n
PX,Ysup
X
n1
inf
Px∈X
n1
Px
/
0
Px
Y
x
X
, 2.8
where X
n1
is any subspace of span {Px : x ∈ X} of dimension ≥ n 1.
2.1. Lower estimate of Bernstein n-width
Consider the extremal problem
x·
p
p
L
r
Dx·
p
p
−→ inf,x· ∈ L
2n
, 2.9
anddenotethevalueofthisproblembyα
p
. Let us show that α ≥ λ
n
, this will imply the desired
lower bound for b
2n−1
.Letx· ∈ L
2n
,then
L
r
Dx·
p
p
2n
i1
Δ
i
2n
i1
b
i
y
i
t
p
dt
2n
i1
Δ
i
|b
i
|
p
|L
r
Dx
n
t|
p
dt
1
2n
2n
i1
|b
i
|
p
, 2.10
and by setting
z
i
· :
1
π
T
G
r
·−τy
i
τdτ, i 1, 2, ,2n, 2.11
we reduce problem 2.9 to the form
μ
j1
a
j
ϕ
j
·
2n
i1
b
i
z
i
·
p
p
1/2n
2n
i1
|b
i
|
p
−→ inf,a
1
, ,a
μ
,b
1
, ,b
2n
∈ R. 2.12
Feng Guo 5
This is a smooth finite-dimensional problem. It has a solution
a
1
, a
μ
, b
1
, ,b
2n
,and,
moreover, b
1
, ,b
2n
/
0. According to the Lagrange multiplier rule, there exists a η ∈ R
such that the derivatives of the function a
1
, ,a
μ
,b
1
, ,b
2n
→ ga
1
, ,a
μ
,b
1
, ,b
2n
ηb
1
b
2
··· b
2n
where g· is the function being minimized in2.12 with respect to
a
1
, ,a
μ
,b
1
, ,b
2n
at the point a
1
, a
μ
, b
1
, ,b
2n
are equal to zero. This leads to the
relations
T
ϕ
j
t
Q
p
x
tdt 0,j 1, ,μ,
2.13
T
z
i
t
Q
p
x
tdt
α
p
2n
Q
p
b
i
,i 1, ,2n,
2.14
where
x·
μ
j1
a
j
ϕ
j
t
2n
i1
b
i
z
i
·.
We remark that ga
1
, ,a
μ
,b
1
, ,b
2n
gda
1
, ,da
μ
,db
1
, ,db
2n
for any d
/
0,
and hence the vector d
a
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,andb
i
0
−1
i
0
1
for some i
0
, 1 ≤ i
0
≤ 2n.
Let
x
2n
t
μ
j1
a
j
ϕ
j
t
2n
i1
−1
i1
z
i
t, 2.15
and x
2n
satisfies 1.7.Leta
a
1
, ,a
2n
and b
1, −1, ,1, −1 ∈ R
2n
. It follows from the
definitions of x
2n
· and x· that
L
r
Dx
2n
t −L
r
Dxt
2n
i1
i
/
i
0
−1
i1
− b
i
χ
i
tL
r
Dx
2n
t −
i − 1π
n
, 2.16
and hence S
c
L
r
Dx
2n
·, L
r
Dx· has at most 2n−2 sign changes. Then, by Rolle’s theorem,
S
c
L
r
Dx
2n
· −L
r
Dx· ≤ 2n − 2. For any a, b ∈ R, signa bsignQ
p
a Q
p
b,therefore
S
c
Q
p
x
2n
· −
Q
p
x·
≤ 2n − 2. 2.17
In addition, since x
2n
is 2π-periodic solution of the linear differential equation
L
r
Dyt−1
r
λ
−p
Q
p
xt,andϕ
j
t ∈ KerL
r
D. Then, by 7, page 94,wehave
T
ϕ
j
t
Q
p
x
tdt 0,j 1, ,μ. 2.18
If we now multiply both sides of 2.15 by Q
p
x
2n
t, and integrate over the interval
Δ
i
, 1 ≤ i ≤ 2n,weget
Δ
i
z
i
t
Q
p
x
2n
tdt −1
i1
Δ
i
|x
2n
t|
p
dt −1
i1
λ
p
2n
2n
. 2.19
Due to
T
z
i
t
Q
p
x
2n
tdt
Δ
i
z
i
t
Q
p
x
2n
tdt.Therefore,wehave
T
z
i
t
Q
p
x
2n
tdt −1
i1
λ
p
2n
2n
,i 1, ,2n. 2.20
6 Journal of Inequalities and Applications
Changing the order of integration and using 2.14 and 2.20,wegetthat
Δ
i
L
r
Dx
2n
t −
i − 1π
n
1
π
T
G
r
t − τ
Q
p
x
2n
τ −
Q
p
x
τ
dτ
dt
T
z
i
t
Q
p
x
2n
t −
Q
p
xt
dt
1
2n
−1
i1
λ
p
2n
− α
p
Q
p
b
i
.
2.21
Denote by f· the factor multiply L
r
Dx
2n
t − i − 1π/n in the integral in the left-hand side
of this equality. If we assume that λ
2n
>α, then we arrive at the relations
sign
Δ
i
L
r
Dx
2n
t −
i − 1π
n
f·dt −1
i1
,i 1, ,2n. 2.22
Suppose for definiteness that L
r
Dx
2n
t − i − 1π/n > 0interiortoΔ
i
,i 1, ,2n.
Then it follows from 2.22 that there are points t
i
∈ Δ
i
such that signft
i
−1
i1
,i
1, ,2n,thatis,S
c
f· ≥ 2n − 1. But f· is periodic, and hence S
c
f· ≥ 2n, therefore,
S
c
L
r
Df· ≥ 2n. Further, L
r
Df·
Q
p
x
2n
t − Q
p
xt,thatis,S
c
Q
p
x
2n
t −
Q
p
xt
≥ 2n.
We have arrived at a contradiction to 2.17, and hence λ
2n
≤ α.Thusb
2n−1
W
p
L
r
; L
p
≥
λ
2n
.
2.2. Upper estimate of Bernstein n-width
Assume the contrary: b
2n−1
W
p
L
r
; L
p
>λ
2n
, 1 <p<∞. Then, by definition, there exists a
linearly independent system of 2n functions L
2n
: span
f
1
, ,f
2n
⊂ L
p
and number γ>λ
2n
such that L
2n
∩ γSL
p
⊆L
r
D, or equivalently,
min
x·∈L
2n
x·
p
L
r
Dx·
p
≥ γ>λ
2n
. 2.23
Let us assign a vector c ∈
R
2n
to each function x· ∈ L
2n
by the following rule:
x· −→ c c
1
, ,c
2n
∈ R
2n
, where x·
2n
j1
c
j
f
j
·. 2.24
Then 2.23 acquires the form
min
c∈R
2n
\{0}
2n
j1
c
j
f
j
·
p
2n
j1
c
j
L
r
Df
j
·
p
≥ γ>λ
2n
. 2.25
Let c
0
0. Consider the sphere S
2n−1
in the space R
2n
with radius 2π,thatis,
S
2n−1
:
c : c c
1
, ,c
2n
∈ R
2n
, c
2n
j1
|c
j
| 2π
. 2.26
Feng Guo 7
To every vector c ∈
R
2n
we assign function ut, c defined by
ut, c
⎧
⎨
⎩
2π
−1/p
sign c
j
, for t ∈
t
k−1
,t
k
,k 1, ,2n,
0, for t t
k
,k 1, ,2n − 1,
2.27
where t
0
0,t
k
k
i1
|c
i
|,k 1, ,2n, and the extended 2π-periodically onto R.
An analog of the Buslaev iteration process 8 is constructed in the following way: the
function xt, c is found as a periodic solution of the linear differential equation L
r
Dx
0
u, then the periodic functions {x
k
t, c}
k∈N
are successively determined from the differential
equations
L
r
Dx
k
t
Q
p
y
k
t,
L
r
Dy
k
t−1
r
μ
−p
k−1
Q
p
x
k−1
t,
2.28
where p
p/p − 1, and the constants {μ
k
: k 0 , ,} are uniquely determined by the
conditions
L
r
Dx
k
p
1,
Q
p
x
k
t ⊥ KerL
r
D,
Q
p
y
k
t ⊥ KerL
r
D. 2.29
By analogy with the reasoning in 8, we can prove the following assertions:
i the iteration procedure 2.28-2.29 is well de fined, the sequences {μ
k
}
k∈N
is
monotone nondecreasing and converge to an eigenvalue λc > 0 of the problem 1.7,
ii the sequence {x
k
·,c}
k∈N
has a subsequence that is convergent to an eigenfunction
x·,c of the problem 1.7,withλcx·,c
p
,
iii for any k ∈
N there exists a c ∈ S
2n−1
such that x
k
·, c has at least 2n zeros
Z
c
x
k
·, c ≥ 2n on T,
iv in the set of spectral pairs λc,x·,c, there exists a pair λc,x·, c such that
S
c
x·, c2N ≥ 2n.
Items i and 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 ∈ S
2n−1
such that
Z
c
x
k
·, c ≥ 2n−1, but since the function x
k
·, c is periodic, we actually have Z
c
x
k
·, c ≥ 2n.
Finally, item iv,byii and iii, which Z
c
x·, c ≥ 2n.Inviewofx·, c zeros are simple,
therefore, S
c
x·, c ≥ 2n.
Since spectral pairs of 1.7 are unique and the Kolmogorov width d
2n
W
p
L
r
; L
q
λ
2n
p, q, L
r
for p ≥ q 5,whenn ≥ N, it follows that
λcλ
2N
d
2N
W
p
L
r
; L
p
≤ d
2n
W
p
L
r
; L
p
λ
2n
. 2.30
Therefore, by virtue of items i, ii,and2.30,weobtain
min
c∈R
2n
\{0}
2n
j1
c
j
f
j
·
p
2n
j1
c
j
L
r
Df
j
·
p
≤
2n
j1
c
j
f
j
·
p
2n
j1
c
j
L
r
Df
j
·
p
≤
x
k
·, c
p
L
r
Dx
k
·, c
p
≤ λcλ
2N
≤ λ
2n
,
2.31
which contradicts 2.25. Hence b
2n−1
W
p
L
r
; L
p
≤ λ
2n
. Thus, the upper bound is proved.
This completes the proof of the theorem.
8 Journal of Inequalities and Applications
Acknowledgments
Project was supported by the Natural Science Foundation of China Grant no. 10671019 and
Scientific Research Fund of Zhejiang Provincial Education Department Grant no. 20070509.
References
1 V. M. Tikhomirov, Some Questions in Approximation Theory, Izdatel’stvo Moskovskogo Universiteta,
Moscow, Russia, 1976.
2 A. Pinkus, n-Widths in Approximation Theory, vol. 7 of Results in Mathematics and Related Areas 3, Springer,
Berlin, Germany, 1985.
3 G. G. Magaril-Il’yaev, “Mean dimension, widths and optimal recovery of Sobolev classes of functions
on a straight line,” Mathematics of the USSR Sbornik, vol. 74, no. 2, pp. 381–403, 1993.
4 A. P. Buslaev, G. G. Magaril-Il’yaev, and N. T’en Nam, “Exact values of Bernstein widths of Sobolev
classes of periodic functions,” Matematicheskie Zametki, vol. 58, no. 1, pp. 139–143, 1995 Russian.
5 S. I. Novikov, “Exact values of widths for some classes of periodic functions,” East Journal on
Approximations, vol. 4, no. 1, pp. 35–54, 1998.
6 N. T. T. Hoa, Optimal quadrature formulae and methods for recovery on function Classds defined by variation
diminishing convolutions, Candidate’s dissertation, Moscow State University, Moscow, Russia, 1985.
7 V. A. Jakubovitch and V. I. Starzhinski, Linear Differential Equations with Periodic Coeflicients and Its
Applications, Nauka, Moscow, Russia, 1972.
8 A. P. Buslaev and V. M. Tikhomirov, “Spectra of nonlinear differential equations and widths of Sobolev
classes,” Mathematics of the USSR Sbornik, vol. 71, no. 2, pp. 427–446, 1992.
9 K. Borsuk, “Drei S
¨
atze
¨
uber die n-dimensionale euklidische Sph
¨
are,” Fundamenta Mathematicae, vol. 20,
pp. 177–190, 1933.
